· preface the international conference on differential and functional differential equa tions has...

FUNCTIONAL IFFE NTIAL E UATIO S

VOLUME 8, 2001 No. 3~4

THE RESEARCHAU'fHORITY THE COLLEGE OF JUDEA & SAMARIA

ARIEL, ISRAEL

The College of Judea and Samaria The Research Authority Executive Director: Dr. Y. Eshel

©All Rights Reserved Printed in Israel ISSN 0793-1786

PREFACE

The International Conference on Differential and Functional Differential Equations has been held in Moscow, Russia on August 16 21, 1999. The conference was organized by Moscow State Aviation Institute, the Steklov Institute of Mathematics of the Russian Academy of Sciences, and Moscow Mathematical Society.

The present volume (vol. 8, NO 1 - 4, 2001) is based on the talks presented at the conference. It includes 35 papers of participants from 10 countries, which were delivered as 45-minute invited lectures and 30-minute lectures.

The conference objective was to discuss classical and new tendencies of development and interconnection of different fields of differential equations. The main topics ranged over differential equations with meromorphic coefficients, nonlinear partial differential equations, nonlinear problems of mathematical physics, geometric and algebraic methods in theory of differential equations. In particular, the proceedings include a generalization of the famous Riemann-Hilbert problem, results on the strong solvability of the N avier-Stokes equations, analysis of attractors for the nonlinear Shri:idinger equations, etc. Along with the above-mentioned topics this volume also contains nonclassical problems for differential equations: nonlocal boundary value problems and functional differential equations with partial derivatives. These new fields have been intensively developing during the two last decades of this century and have important applications to elasticity theory, plasma theory, theory of Feller semigroups arising in biophysics, and to nonlinear optics.

We hope that this volume will be interesting for a wide spectrum of specialists and graduate students in differential equations, functional differential equations, and mathematical physics.

The editors are greatful to Professor Alexander Scubachevskii for the great help provided by him in preparing the volume for publication.

M. Drakhlin,

E. Litsyn.

TABLE OF CONTENTS

V.A. Golubeva. On the Riemann-Hilbert Problem in a Class of Knizhnik-Zamolodchikov Equations.

A.M. Il'in and S.V.Zakharov. On the Influence of small

241

Dissipation on the Evolution of Weak Discontinuities. 257

Y. Ishikawa. Density Estimate in Small Time for Jump Processes with Singular Levy Measures and its Support Property. 273

G.A. Kamenskii. Moving Boundary Problems for Nonlocal Functionals. 287

Z. Kamont. Initial Problems for Hyperbolic Functional Differential Equations with Unbounded Delay. 297

V .L. Kamynin. Passage to the Limit in Quasilinear Parabolic Integro-Differential Equations on a Plane 311

V.G. Kurbatov Some Algebras of Operators Majorized by a Convolution. 323

V. P Lexin. Monodromy for the Knizhnik-Zamolodchikov (KZ) Equations of the En-Type and Accompanying Algebraic Structures. 335

L. Moschini, A. Tesei and S.I. Pohozaev. On a Class of Nonlinear Dirichlet Problems with First Order Terms. 345

A.B. Myravnik. Fourier-Bessel Transformation of Compactly Supported non-Negative Functions and Estimates of Solutions of Singular Differential Equations. 353

239

240

S. Nikolskii. Boundary Value Problem for Algebraic Pol)nomials. 365

V.A. Plotnikov and P.I. Rashkov. Averaging in Differential Equations with Hukuhara Derivative and Delay. 371

J. Qing. A Fourth Order PDE and its Application in Conformal Geometry. 383

L.E. Rossovskii. On the Boundary Value Problem for the Elliptic Functional-Differential Equation with Contractions. 395

A.L. Skubachevskii and R.V. Shamin. The Mixed Boundary Value Problem for Parabolic Differential-Difference Equation. 407

S.A. Stepin. On the Asymptotic Behavior of Solutions to Second Order Ordinary Differential Equations. 425

V.V. Vlasov. On Spectral Problems Arizing in the Theory of Functional Differential Equations. 435

N.D. Vvedenskaya and Yu.M. Suhov. Differential Equations Arising in Queuing Theory. 44 7

G.A. Yosifian. Homogenization of Problems with Friction for Elastic Bodies with Rugged Boundaries. 459

FUNCTIONAL DIFFERENTIAL EQUATIONS

VOLUME 8 2001, NO 3-4 PP. 241-256

ON THE RIEMANN-HILBERT PROBLEM IN A CLASS OF KNIZHNIK-ZAMOLODCHIKOV EQUATIONS*

V.A. GOLUBEVA t

Abstract. The results connected with the restricted Riemann-Hilbert problem in a class of Knizhnik-Zamolodchikov equations for different root systems are reviewed. In particular, the case of the root system Bn is considered in detail. Problems for other root systems are stated.

Introduction. In 1857 B. Riemann stated the problem of recovery of an ordinary differential equation of hypergeometric type if the monodromy of its solutions is given. More exactly, the problem was the following: let A = { ab ... ,an} be a set of points in the complex plane C and p : 1r1 (C \A, z0 ) -+ G L(p, C) be a given representation p; does there exist a differential equation of Fuchsian type with the set of singular points A, whose monodromy realizes the given representation?

For n = 3 (the case ofhypergeometric equation), this problem was solved by Riemann. If n > 3, the number of parameters of data is less than the number of parameters of the reconstructed differential equation. And in 1901 D. Hilbert stated an analogue of the problem for systems of ordinary differential equations of the first order and of the Fuchsian type (here the problem of deficiency of parameters is absent). This problem called the Riemann-Hilbert problem (below RHP) was solved by A.A. Bolibrukh in 1989.

Naturally, it would be interesting to generalize this problem to higher dimensions. A higher-dimensional analogue of the Riemann-Hilbert problem was announced first by I. M. Gelfand in the 1960s, then by O.S. Parasyuk, and later one of the versions of the generalized Riemann-Hilbert problem

• This work was supported by INTAS, Grant N 97-1664 t All-Russian Institute Sci-Tech Info, Usievich 20, Moscow 125219 Russia

241

242 V.A. GOLUBEVA

was stated in 1965 by Italian physicist T. Regge (see [1)) who noted that the analytic properties of the Feynman integrals are similar to the properties of hypergeometric functions. He conjectured that the Feynman integrals must satisfy some systems of differential equations of hypergeometric type (see [2)). In the 1970s it was not even clear what are higher-dimensional differential equations of hypergeometric type.

In 1969-70 the idea of higher-dimensional generalization of the RHP was announced by two mathematiciens: R. Gerard [3] and P. Deligne [4]. The problem was stated in the following form.

Let L = U;L;, degL; = n;, be some reducible algebraic variety in <CJP>n and p : 1r1 (<CJP>n \ L, z0 ) -+ GL(p, C) be a given representation. Does there exist a Pfaff system of Fnchsian type

the monodromy of which realizes the representation? The first results in its solution were obtained at the end of the 70s and

in the 80s with some simplifying conditions for the variety L providing local commutativity of the fundamental group 1r1 (IClP'n \ L, z0 ). In 1977 A.A. Bolibrukh obtained a solution of the local RHP, i.e. in the neighbourhood of the origin of C'. This approach permitted him to appreciate the role of valuations in the solvability of the problem and to understand the interplay of local solutions of the !-dimensional RHP in the global picture, which resulted in a negative answer to Hilbert's question [5].

A series of results on higher-dimensional Riemann-Hilbert problem was also obtained by V. Golubeva, M. Ohtsuki, R. Hain, M. Kita, V. Leksin and others but the case of non commutative fundamental group appeared very complicated for investigations. For this reason, it would be natural to restrict a class of singular varieties of the resulting differential equations. For example, the divisor L was supposed to be a set of hyperplanes in C' having some groups of symmetries, in particular, the well-known hyperplane arrangements associated to different root systems. The Fnchsian systems corresponding to these divisors were constructed recently by I. Cherednik, A. Leibman, A. Matsuo, V.Golubeva and V. Leksin (see [6)). These are the socalled generalized Knizhnik-Zamolodchikov (below KZ) equations. It apF>ears that these systems can be considered as higher-dimensional generalizations of hypergeometric differential equations.

It is necessary to note that the statement of the higher-dimensional problem as a simple generalization of the one-dimensional one is not suffi-

ON THE RIEMANN-HILBERT PROBLEM 243

ciently satisfactory. Indeed, in one-dimensional case the fundamental group 1r1 (<C\A, z0 ), as a rule, has a very simple structure. It is a free group with a finite number of generators. In the higher-dimensional case the structure of the fundamental groups 1r1 (<C" \ L, z0) is considerably more complex. Although these fundamental groups are known for a number of arrangements, their representations are not investigated sufficiently exhaustively. Since the final equations that are questionned in the statement of the higher-dimensional problem are known, the RHP has a slightly different form.

More exactly, the statement of the RHP in a class of KZ equations is the following.

Let R be a given root system and L = UiLi be the corresponding hyperplane arrangement, i.e. the set of mirrors of the corresponding Weyl group WR.

The problem. a) Characterize the fundamental group 1r1 (<C" \ L, z0 );

b) keeping in mind the generalized KZ equation corresponding to a given root system, characterize the class of representations

p: 1!'1 (<CIP'n \ L, zo) -+ GL(p, C)

which could be realized as the monodromy representations of the generalized KZ equation associated to a given root system.

Below we give a review of all results we know concerning complete or particular solutions of the restricted RHP in a class of KZ equations for different root systems.

1. The Kohno Theorem for the Root System An-!• In this section we give a short review of a result ofT. Kohno on solving the problem of recovery for a system of equations of Fuchsian type in <C", when a representation of the fundamental group of the complement in <C" to the singular divisor of the system to G L( n, q is defined so that it factorizes through the Heeke algebra [7]. Further, in the next section we will consider a generalization of this result to the case of the KZ equation of the Bn type.

Let An-! be a given root system in <C", and WAn-l be the corresponding Weyl group. Let H' be the set of reflection hyperplanes (the so-called hyperplane arrangement) of the Weyl group, H' = UiHij, where

Hij = {(z1, ... , Zn): Zi- Zj = 0,1 :S i < j :S n},

for the root system An, and let Xn be its complement in <C", that is,

Xn = ((z1, ... , Zn) E <C"; Zi f. Zj for if. j).

244 V.A. GOLUBEVA

The fundamental group 1!'1 (Xn, z0 ), z0 E Xn, is a pure braid group Pn. It will be denoted below P(An-d· For the Artin braid group Bn, we will use below the notation BAn-1·

The corresponding KZ equation has the form df = nj, where

nAn-1 = L nij log(z;- Zj), i<j

and f is a p-dimensional column of holomorphic functions on the universal covering Xn.

The integrability conditions of this equation are consequences of the relations dO, = 0 and n A n = 0. Written in terms of coefficients n;; of the 1 form n, these conditions have the following form:

[n;3, nik + n3k] = [n3k,n;; + n;k] = o for i < j < k,

(n;3, nkt] = o for {i,j} n {k, I}= 0,

These relations are called the infinitesimal relations of the pure braid group of the An-1 type.

~' The following well-known theorem holds.

THEOREM 1 (V. Golubeva [8],[9], T. Kohno [7] and R. Hain [10]). Let

PAn-1 : 'll'1(Xn) -+ GL(p, C)

be a linear representation such that p( 1) is close to the identity representation for all generators ri E 'll'1(Xn)· Then there exist constant matrices n;J, 1 :::; i < j :::; n close to 0 that satisfy the infinitesimal braid relations of the An-1 type and are such that the monodromy of the connection nA._1 coincides with a given representation PA.- 1 •

Now we consider a KZ equation that is invariant with respect to the action of the symmetric group Sn· We assume that Sn acts on <C" by permutation of coordinates, and it acts on the coefficients of the equation according to the rule

Then the monodromy of the KZ equation defines the representation

PKZ : B(An-1) -+ GL(p, iC)

of the braid group B(An-d in the matrix algebra.

In this case, the following Theorem 2 holds. Let q be a nonzero complex number. The Iwahori-Hecke algebra H(q, n)

is the associative algebra (over CC) generated by g1, ... , gn-1 with the relations

gf =(q-l)g;+q, g;gi+1gi = g;+lgigi+b

g;gj = gjgi, li - j I :2: 2.

THEOREM 2 (T. Kohno [7]). Let

PB(An-l)(q) : B(An-l) --t GL(p, C)

be a linear representation factoring through the Heeke algebra H(q, n). If we assume that q is sufficiently close to 1, then there exist p x p-matrices O;J ( q), 1 ~ i < j ~ n, satisfying the infinitesimal braid relations, such that the monodromy of the connection nAn-! (q) is a restriction of PB(An-1 )(q) to the pure braid group P(An-1)·

In the next section we give a generalization of this theorem to the case where the hyperplane arrangement H is the set of mirrors for the Weyl group of the root system Bn.

2. On the RHP in a Class of the KZ Equations of the Bn Type. We consider now the case of the root system Bn in en. Let H be the union of reflection hyperplanes of the Weyl group W Bn,

where

H;j = {z;- ZJ = 0,1 ~ i < j ~ n}, H;j = {z; + ZJ = 0,1 ~ i < j ~ n}, HZ = {zk = 0,1 ~ k ~ n}.

Further, let Yn be its complement in en, Yn = en \ H. The fundamental group 1T1(Yn,Yo), y0 E Yn, is the generalized pure braid group of the Bn type that was first described by A. Leibman [11]. Further it will be denoted by P(Bn)· The Brieskorn braid group of the Bn type will be denoted by B(Bn)· We consider the matrix-valued 1-form

fJBn = I;(fJ;jdlog(z;- Zj)) + fJ~dlog(Z; + Zj)) i<j

n +I; O?dlogz;,

i=l

246 V.A. GOLUBEVA

over Yn with a constant p x p-matrix coefficients !}~ and !12. The KZ equation of the Bn type has the form

dlJ!(z) = !}Bn IJ!(z),

where lJ! ( z) is a p-dimensional column of holomorphic functions on a universal covering Yn.

It is assumed that the connection !}Bn is integrable, that is, the relations d!l = 0 and !} 1\!} = 0 hold. These integrability conditions are equivalent to the following relations:

[nt,% + njkl = o, [n;;, n;t + njkl = o, [nu,, n;; + !ljkl = o

and similar relations with cyclic permutations of the subscripts i, j and k, and also the relations

[nt, n;:; + n? + n11 = o, [!li, nt + nr + nJ] = 0, [!lij + n;:; + nr' nJ] = 0,

[nii; nttJ = o, [!lij' nz] = 0.

The last two relations are considered for the subscripts with distinct sets i, j and k,l.

These relations are called the infinitesimal relations of the pure braid group P(Bn)·

The set of hyperplanes H is a ramification divisor of the fundamental matrix IJ!(y) of the KZ equation of the Bn type satisfying the condition IJ!(yo) =I, Yo E Yn.

The monodromy of the KZ equation is defined by the series of iterated integrals (see K.-T. Chen [12])

p('y) = 1 + J!} + J !}!} + ... + J nn + ... ' ~ ~ ~

where f nn is an iterated integral over the loop 'Y E 11'1 (Yn)· ~

The monodromy representation of the KZ equation defines a represen-tation of P(Bn) to the group of authomorphisms of the vector space V, dimV =p,

PKZsn : ?rr(Yn)--+ GL(p, C).

The restricted RHP for the case of the root system Bn is the following: A representation

is given. Does there exist a KZ equation of the Bn type whose monodromy realizes this representation?

It was shown (see [13]) that the monodromy of the formal connection of the variables Xr, ... , X;t', XZ

3Bn = L;(Xijdlog(zi- Zj)) + X;"jdlog(zi + Zj)) i<j

n +I: xpd log zi,

i=l

given by the Chen series defines a faithful representation of the braid group of the Bn type in the quotient of the algebra of formal power series of variables Xij, Xi} and Xp by the ideal generated by the integrability conditions 3Bn A 3Bn = 0.

We can state the following theorem.

THEOREM 3. Let PBn : PBn -+ GL(p,CC) be a linear representation such that the norm lip(!)- Ill is sufficiently small for all generators of PBn.

Then there exist constant matrices nt, n;:;, and nz such that they satisfy the infinitesimal relations of the generalized pure braids P(Bn), and the monodromy of the connection nBn coincides with a given representation PBn.

For the proof of the theorem, see [13]. Now we will give some subtle sufficient conditions for the representation

PBn, under which it can be realized as the monodromy of some generalized KZ equation of the Bn type. These representations will be connected with some Heeke algebras of the Bn type denoted HBn (q, Q) (see [14]).

An algebra H 8 n (q, Q) is an associative algebra with unity over the field <C generated by x0 , x1 , ... , Xn-! satisfying the relations

x[ = (q- 1)xi + q, 1:::; i:::; n- 1, x5=(Q-1)xo+Q,

XiXjXi = XjXiXj, li- Jl = 1, i,j?: 1, XoX 1XoX 1 = X 1XoX 1x 0 ,

XiXj = XjXi 1 li- Jl ?: 2.

Here q and Q are some real constants. The following theorem holds.

248 V.A. GOLUBEVA

THEOREM 4. Let

PB(Bn) : B(Bn)--? GL(p, C)

be a linear representation factoring through the H ecke algebra H Bn. Let q and Q be two numbers sufficiently close to unity. Then there exist p x p matrices nt)(q,Q), 1:::; i < j:::; n, n~(q,Q), 1:::; k:::; n, satisfying the infinitesimal relations of the pure braids P(Bn) such that the monodromy PKz8 • of the connection nB. is a restriction of PB(Bn) to the pure braid group P(Bn).

REMARK 1. The Heeke algebra used above depends on two parameters. As we will see in the next section, such number of parameters is natural to introduce in the 1-form defining the KZ equation of the Bn type.

REMARK 2. Theorems 2 and 4 can be stated and proved for other factorizations of the given representation, for example, for the factorization through the Birman-Wenzl algebras of the An and Bn type (see, for example, [26],[38]).

More precisely the properties of the monodromy for the KZ equation of the Bn type will be characterized in the next section.

3. On the Quasi-Bialgebra Structures Associated with the Representations of Braid Groups and the Generalizations of the Drinfeld-Kohno Theorem. In this section we give a brief sketch of the results of V. Drinfeld and T. Kohno devoted to the investigation of connection between the quasi-bialgebra structures associated to the braid group representation and the monodromy representations of the corresponding KZ equation of the An-1 type. Then we give a generalization of the principal ingredients of this theory to the case of the root system Bn. In conclusion, we will consider results concerning the cases of other root systems.

3.1. Case of the root system An_1• A quasi-bialgebra is a quasi commutative and quasi coassociative bialgebra A with structure

where J1 is multiplication, ~ is comultiplication, 'fJ is unit, c is counit. Moreover, this structure includes two additional elements R and <I? defined in the following way.

Let r be the map T: A 02 --? A 02 such that a®b--? b®a and ~op = T·~. Further, we have

~OV(a) = R~(a)R-1 ,

~ 0 id)~(a) = <.P(id0 ~)~(a)w- 1 .

ON THE RIEMANN-HILBERT PROBLEM

The quasi-bialgebra structure satisfies the following axioms:

where

1. ~IRA= .p~2. R;{. (<P;f2)-l. R~3. <P;f3,

A R - (-"231)-1 Rl3 "-213 Rl2 ("-132)-1 '-'2 A- '*'A . A • ..,A • A. '*'A >

3. ~a(<PA) · ~t(<l>A) = (J 0 <l>A)~2(<PA)(<PA 0 /),

5. CJ(<PA) = c2(<l>A) = c3(<l>A) = 1,

c;(R) = 101, e;(<P) = 10101.

c;=£01, 10£, £0101, 10£01, 1010£,

c is the counit of A. The quasi-bialgebra structure

permits to define the representation

249

By a star is denoted the set of invertible elements of the corresponding object.

Let now g be a semisimple Lie algebra, U(g) be an universal enveloping algebra, Uh(g) be a quantum universal enveloping algebra, (U(g)[[h]J, J.L, ~)be the quantum Drinfeld-Jimbo deformation of U(g), t be the Belavin-Drinfeld tensor

where t E g 0 g, c is the Casimir element in U(g). Further the notations t12 = t 0 1, t2a = 1 0 t will be used.

The KZ equation of the An-J type with parameter h has the form

df = h(:~::>ijdlog(z;- zjl)J, i<j

where t;J is the image of the element t under natural inclusion

250 V.A. GOLUBEVA

It is known that the Drinfeld-Kohno theorem [15],[16],[17] (see also [18]) gives a characterization of the monodromy PKZ of the classical (i.e. associated to the root system An) KZ equation by the quasi-bialgebra structure

where RKz = ehtt212 is the R-matrix, if>Kz is the Drinfeld associator and J1 and A are the multiplication and comultiplication operators, respectively. This description is universal in the following sense: if the representation of a braid group Ph

is constructed by means of the quasi-bialgebra structure

on the quantum universal enveloping algebra of Drinfeld-Jimbo Uh[g] with ~. non-trivial R-matrix Rh and trivial Drinfeld associator if>= 10101, then by

the Drinfeld-Kohno theorem these representations PR• and PKz are equivalent (see [18], 19.4.1). In other words, there exists a <C[[h]J-linear isomorphism u of (Uh(g))®n and (U(g)[[h]])®n such that

,,

PKz(g) = U · PRh (g)· U-1

for all elements g of the braid group B(An-d· By the Drinfeld theorem ([15], [16], and [18], theorem 19.4.3), this equiv

alence of representations is defined by a <C[[h]J-linear isomorphism a between the quasi-bialgebra A9,t, twisted by some transformation F(h) defined as formal series F(h) = 1 + fh + fh2 + ... ,and the quasi-bialgebra Ah

3.2. Case of the root system Bn. Consider the generalized KZ equation with two parameters h1 and h2 associated to the root system Bn

The coefficients tij, tt, t~ are defined by the elements r E U(g)02 , t 0 E U(g) and by the Weyl-Chevalley automorphism o-w : U(g) --t U(g).

Let t- be the Be!avin-Drinfeld tensor

n

r = L ea0La+ L9ijhi0hj, o:ER+ i,j=l

where R+ denotes the set of positive roots of the root system R, and

t0 = L: eae-a + e_"e"

o:ER+

the Leibman element [19]. Let further t+ = (o-w 01)t. Besides, by tij, tt, t? we will denote the images of the elements r, t+, t 0 under natural inclusions

U(g)®2 --t U(g)®n

on i-th and j-th factors for t± and on i-th factor for t0 , respectively. We suppose that the integrability conditions for the 1-form fl at the right hand of the KZ equation of the Bn type are fulfilled.

Define now the quasi-bialgebra structure of the Bn type on an algebra A. It is characterized by the following data:

where rJ is a unit, cis a counit, RA, A were described above, and R8 E A, 8 E A02 are some new elements, which at this moment have no the algebraic definition. However, it is clear that if the element RA is responsible for the solution of the Yang-Baxter equation [19], R8 is responsible for the solution of the reflection equations (see [20]-[23]). In addition to the axioms 1,3,5 written above, we give the following axioms containing the elements R8 , 8 .

We have

2. A(RB) =<I![/ . (RB 0 I). q,B. R~2 . }/ . (RB 0 I). <l!B,

4. Al(<PB) = <1!:41• (B 0 I). <I! A, A2B = <l!B 01,

6. c(RB) = 1, (c 0l)<Pn = (10 c)B = 10 1.

See a geometric approach to these axioms in [13], [24].

252 V.A. GOLUBEVA

Let now g be a semisimple Lie algebra and Uh(g), where h = (h1, h2), be the topological algebra over IC[[ h1 , h2JJ. We consider topological quasibialgebras of the Bn type of the form

Any such quasi-bialgebra is connected with a representation of the braid group B(Bn)· In particular, the quasi-bialgebra structure of the Bn type with <P A = 1 ® 1 ® 1 and <P 8 = 1 ® 1 defines the elements

We consider now the quasi-bialgebra structure connected with the KZ equation of the Bn type. It is characterized by the following data:

BB,Kz = ( U(g)[[h1, h2Jl,Jt, ~,c, 1], RA,KZ, RB,Kz, <PA,Kz, <PB,KZ)

RA,KZ = eh,t- E (U(g)[[hb h2]])02, RB,Kz = eh2t0 E U(g)[[hb h2]],

<PA,KZ E (U(g)[(hl, h2]])03, <PB,KZ E (U(g)[[hb h2]])02.

We define now the twisting of the bialgebra of the Bn type on U(g)[[h1,h2]]. It is defined by means of two series FA E U(g)[[h1,h2]]02

and FB E U(g)[[hb h2]J. The twisted elements are obtained in the following way:

- -1 - -1 RB = FBRBFB , RA = FARAFA ,

•h = (~FB)<PB(~FB)-1, <PA = (1 ® FA)~2FA<PA~l(F,4 1 )(FA ® 1),

Li(a) = FA~(a)Fi\ jl(a ®b)= ~t(FA(a ® b)F,41)

The twisted set of objects also satisfies axioms 1-6. The following theorems holds.

THEOREM 5. For any semisimple Lie algebra g there exist universal R-matrices RA,h, RB,h of the corresponding quasi-bialgebra structure Bh of the Bn type with trivial associators <P A = 1 ® 1 ® 1 and <P 8 = 1 ® 1 satisfying the relation


THEOREM 6. There exist two twisting series F(h1o h2) =FA, FB, for which the following isomorphism of the quasi-bialgebras holds:

BB,KZ

Using the definition of twisting, we can show that two isomorphic (i.e. connected by twist) quasi-bialgebras define equivalent representations. We obtain

THEOREM 7. Any representation PRA,h,Re.h of the braid group of B(Bn) constructed by means of a quasi-bialgebra structure Bh(Bn) of the En-type is equivalent to the monodromy representation of the KZ equation of Bn type

The arguments in the proofs of the theorems are similar to the arguments of the proof of the Drinfeld-Kohno theorem for the An case (see [18], ch. 19). But the proofs are too long to give them in this short pap~r.

REMARKS.

1. As it is obvious from the above text, for a semisimple Lie algebra there is no algebraic definition of the matrix R8 (the reflection matrix) analogous to the definition of the matrix RA as twisting operator for 6 and ,6. op given in Section 3.1.

2. As we have seen, the generalization of the Drinfeld-Kohno theory to root systems different from the root system An-I is connected with the problem of existence and rigidity of two- (and multi-) parametric deformations of topological quasi-bialgebras. Although multiparametric deformations of some algebraic structures were considered in a series of papers (see, for example, [25]-[32]), the algebraic structures used in the Drinfeld-Kohno theory were not considered. The sufficient conditions for the existence of two- (and multi-) parametric deformations of quasi-bialgebras are not known. Most of the papers [25]-[32] are devoted to the multiparametric solutions of the Yang-Baxter equations, but we do not know any papers in which, in addition to the Yang-Baxter equations, multiparametric solutions of reflection equations are investigated.

254 V.A. GOLUBEVA

3.3. Other root systems. There are a few papers which could be considered as related to the problems considered above and treat root systems different from An, Bn. Here we consider only the root systems Dn and G2.

The case of the root system Dn. The structure of the fundamental group of the complement to the set of the reflection hyperplanes of the Weyl group WBn, the so-called pure braid group P(Bn), was studied in [33]. Problems closely connected with the RHP were considered in [34]. Some monodromy representations of the generalized KZ equations of the Bn type were given in [19]. But the corresponding quasi-bialgebra structures was not investigated.

The case of the root system G2 • We only Jist some dispersed facts that could be used in the investigations of the corresponding algebraic structures. At first, the fundamental group of the complement to the corresponding set of mirrors is easy calculable. The representations of the corresponding Lie algebras and even their geometric treatment, and also the corresponding KZ equations are known (see, for example, [35]-[38]).

REFERENCES

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[9) V.A. Golubeva, On systems with regular singularities and their solutions, Math. USSR Izvestiya, 27, 1 (1986), 27-38.

[10) R. Hain, On generalization of Hilbert's 21st problem, Ann. Scient. Ec. Norm. Sup., 4-me ser., 19 (1986), 609-627.

[11) A. Leibman, Fiber bundles with degenerations and their applications to computing fundamental group, Geom. Dedicata, 48 (1993), 93-126.


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[14] T. Tom Dieck, Knotentheorie und Wurzelsysteme, I, II, Mathematica Gottingesis, 21, 44 (1993).

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groups, Preprint. Nagoya University, 1991. [18] Ch. Kassel, Quantum groups, Graduated texts in Math, 155, Springer, 1995. [19] A. Leibman, Some monodromy of generalized braid groups, Comm. Math. Phys.,

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of reflection equations and their Yang-Baxterization, J. Phys. A: Math. Gen., 26 (1993), 5953-5958.

[22] R. Hiiring-Oldenburg, New solutions of reflection equation from type B BMW algebras, J. Phys. A: Math. Gen., 29 (1996), 5945-5948.

[23] D.I. Gurevich, P.N. Pyatov, and P.A. Saponov, Heeke symmetries and characteristic relations on reflection equation algebras, J. Phys. A: Math. Gen., 41 (1997), 255-264.

[24] V.P. Leksin, Monodromy of the KZ equations of the Bn type and accompaniing algebraic structures - in this volume.

[25] M. Takeuchi, A two-parameter quantization of GL(n), Proc. Japan Acad., SerA, 66 (1990), 112-114.

[26] A. Schirrmacher, Multipararneter R-matrices and their quantum groups, J. Phys. A: Math. Gen., 24 (1991), 1249-1258.

[27] Fei Shao-Ming, Guo Han-Ying, and Shi He, Multiparameter solutions of the YangBaxter equation, J. Phys. A: Math. Gen., 25 (1992), 2711-2720.

[28] C. Burdik and P.J. Hellinger, The universal R-rnatrix and the Yang-Baxter equation with parameters, J. Phys. A: Math. Gen., 25 (1992), 1023-1027.

[29] R.B. Zhang, Multi parameter dependent solutions of the Yang-Baxter equation, J. Phys. A Math. Gen., 24 (1991), 535-541.

[30] A. Aghamohamrnadi, V. Karirnipour, and S. Rouhani, The rnultiparametric nonstandard deformation of An-!, J. Phys. A Math. Gen., 26 (1993), 75-82.

[31] A. Aghamoharnrnadi, V. Karimipour, and A.R. Nezami, Non-standard deformation of Bn series, J Phys. A Math. Gen., 27 (1994), 1609-1616.

[32] Mai Zhong-Qi and Dai An-Ying, A new solution of the Yang-Baxter equation related to the adjoint representation of UqB2, J. Phys. A Math. Gen., 27 (1994), 1999-2009.

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(1993), 431-451. [36] Ge Mo-Lin, Wang Lu-Yu, and X.P. Kong, Yang-Baxterization of the braid group

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VOLUME 8 2001, NO 3-4 PP. 257-271

ON THE INFLUENCE OF SMALL DISSIPATION ON THE EVOLUTION OF WEAK DISCONTINUITIES '

A.M. IL'IN AND S.V. ZAKHAROV t

1. Introduction. We consider the following initial value problem

(1.1)

(1.2) u(x, -1) = u(x) = -(x + ax2) 8( -x) (1 + q(x)), x E JR,

where E: > 0, a > 0, 8 is the Heaviside function, and the function q E coo (JR) is such that u is bounded and, besides, supp q lies outside a sufficiently large neighbourhood of zero. Assume that 0, <p11 (0) = 1. Then there exists a unique solution u(x, t; c) E C00 (S) n C(S) of problem (1.1), (1.2), where s = {(x, t) : X E JR, -1 < t < T}.

By u0 (x, t), we denote the solution of the degenerate problem, i.e. problem (1.1), (1.2) forE:= 0. The equation for characteristics of the degenerate equation has the form

(1.3) x = y + 'P'(u(y)) (t + 1),

where y is constant on characteristics. Suppose that the function q(x) in (1.2) is such that the characteristics do not intersect for t < 0. Therefore the solution u0(.T, t) = u(y) is continuous in the strip -1 ~ t < 0, where the

' This work was supported by the Russian Foundation for Basic Research grants No.96-15-96241, 99-01-00139.

t Institute for Mathematics and Mechanics, Ural Branch, RAS, 620219 Ekaterinburg 16, Russia

257

258 A.M. IL'IN AND S.V. ZAKHAROV

variable y is expressed in terms of x and t by equation (1.3). Condition (1.2) and equation (1.3) imply that

lim ouo(x, t) = ~, x->-0 OX t

lim ouo(x, t) = 0. x->+0 OX

Thus the solution of the degenerate problem has a weak discontinuity on the line f._= {(x,t): x = 0, -1 ~ t < 0}.

If t ;:, 0, the characteristics intersect. In this situation called gradient catastrophe, the degenerate problem admits no smooth solution defined up to positive values oft. However, there exists a generalized solution, which is a piecewise smooth function with a smooth discontinuity curve f.+ = { (x, t) : x = s(t), t;:, 0}.

The asymptotics of the solution u(x, t; c) is fairly complicated in the vicinity of the origin, where the shock wave arises.

In this paper we study the asymptotics of the solution u(x, t; c) fort ~ 0, which has to be known to construct a uniform expansion in the strip S (T > 0).

THEOREM 1.1. An outer expansion of the solution of problem (1.1), (1.2) can be constructed in the form of the series

00

(1.4) U = L cnun(x, t), n=O

where u0 (x, t) = u(y). For n;:, 1 and sufficiently small y < 0,

n 3n-s-1

(1.5) Un(x, t) = 2.:::In8 w(y, t) L [w(y, t)tPi[>p(Y), s=O p=l

ox(y, t) . where w(y, t) =

0 , and i[>P are some smooth functwns.

y

The proof can be found in [1, Chapter VI, Th. 3.2].

2. Boundary Layer Near the Weak Discontinuity Line. An expansion of the solution u(x, t; c) in a neighbourhood of f._ is sought in the form of the series (see [2])

00

(2.1) V = L ckl2vk((, t), k=l

ON THE EVOLUTION OF WEAK DISCONTINUITIES 259

where ( = xc112 is a new variable. Substituting series (2.1) in equation (1.1) and condition (1.2), we obtain the recurrence system of initial value problems

(2.2)

(2.3)

(2.4)

- 1 k-1 k+l ,<,l(o) q where J(v1, ... ,vk-1)- -2 2::VJvk+1-j- I: ! I: rr Vi,, k ~ 3. The

j=2 q=3 q L;i,=k+1 s=1

solution of (2.2) is found by the Hopf-Cole transformation

(2.5)

The function w((, t) is the solution of the following initial value problem

at w((,-1) = exp [ec-o~]. ()([!

The latter has the form

(2.6) {

0 1 ((-ry)2 ry2

w((, t) = 2Jrr(t + 1) _£ exp [ 4(t + 1) + 4] dry+

((- ry)2 00 }

+! exp [- 4(t + 1)] dry .

Formulae (2.5) and (2.6) allow to find the asymptotics of v1((, t) as t -c> -0 for any (. This asymptotics is of a fairly complicated nature, which essentially depends on ratio of the variables of ( and t. To formulate rigorous statements, it is convenient to consider the following domains: Do = { ( (, t) : 1(1 < ltl', -1 < t < o}, D_ = {((, t) : ( < -w, -1 < t < o}, D+ = {((, t) : ( > IW, -1 < t ( 0}. Numbers 5, p E lR are such that 0 < 5 < p < 1/2. Then the intersections D_. nDo and D0 nD+ are nonempty.


THEOREM 2.1. As t-+ -0, the following asymptotic representations hold

(2.7) v!((,t)=t+O([W 112 exp{-~[t[2P-l}), ((,t)ED_,

N

(2.9) v1 ((, t) = L [t[mS1,m(() + 0 ([t[N(!-2p)+l-p), ((, t) ED+, m=O

where N) 0, R1,0,m E 0 00 (lR), S1,m E 0 00 (0, +oo).

Proof. a) Formula (2.6) implies that if ((, t) E D_, then the relations hold as t -+ -0:

(2.10) 1¥((, t) = )=t exp (- ~:) + 0 ([WP),

(2.11) 81¥((, t) = _.f_ {-1- exp (-e) + 0 (IWP)}. 8( 2t A 4t

Combining (2.10), (2.11) and (2.5), we get formula (2.7). b) Now let((, t) E D0 • Representing formula (2.6) in terms of the self

similar variable {) = -d=t with subsequent expansion of limits of integration in Taylor series, we conclude that the following relations hold as t -+ -0:

1 (2.12) w((,t) = 2AA(1J) +

N-!

~ + L [t[n+l/2 P2n+! ( {)) + 0 ([t[(2N+!)o) ' n=O

( ) 8w((,t) Zo(1J) 1 ~lin I () ( 2No) 2.13 8( = 4 tA(1J) + 2 ~ t P2n+! {) +0 [t[ ,

where

00

A({)) = v; e-11' [ J e-v' dy r, 2 Z0 (1J) = -2{} + VifA(1J),

II


and Pk('!9) are some polynomials of degree k. Combining (2.12), (2.13) and (2.5), we get formula (2.8).

c) Finally, let ((, t) ED+. Then formula (2.6) implies that the following relations hold as t --+ -0:

(2.14) IV((, t) = jf~ + exp (-~) ~ tmp~';;;;:~() + 0 (1t1N(1- 2Pl+P),

(2.15)

81V((, t) = exp (- (2) {--1- + ~ tmP4m(() + O (1t1N(1-2p)+2p)}

8( 4 Vif(2 ~ (2m+2 '

( 2) (/2 2 . . where D(() = exp -~ + ( J e-v dy. Combmmg (2.14), (2.15) and (2.5),

-00

we get formula (2.9). D

Next terms of expansion (2.1) satisfy linear PDEs. Since the asymptotics of these terms is studied in an almost identical manner, we present detailed proofs for the function v2 ( (, t) only.

Everywhere below we use the notations

1 ( rP) G(ry,s) = --exp -- . 2,fir8 4s

THEOREM 2.2. As t--+ -0, the asymptotic representation holds:

(2.16) 112((, t) = vg((, t) + 0 (It!'' exp {- ~ wp-1}), ((, t) ED_,

where a > 0 v•(r t) = £.(':- (2b + 'P"'(0);-2) l- rn111 (0)~- ~ and b = ' 2 ":tl t3 2 ":. t2 r t t'

a- cp111 (0)/2.

Proof. From (2.3) it follows that the difference r((, t) = v2 ((, t) -vg((, t) is the solution of the initial value problem

(2.17) £r = -cp111 (0) ((p + p2

+ vf 8P) t2 2t 2 8(

8(11~p)

8(

where p((, t) = 111((, t)-% = O(ltl-112exp { -~wr-l}) in the domain D_. The function r((, t) is given by the expression

8 a 3 8 g( (, t) {

00 } (2.18) r((, t) = 8( 3W((, t) [ 7] G(ry- (, 1 + t) dr1 + 8( {IV((, t)},

;_.


where t 00 ~

(2.19) g((,t)= j j G(ry-(,t-s)>T!(ry,s)JF(y,s)dydryds, -1-oo -00

with F being the right-hand side of the equation for r. Using Theorem 2.1, we conclude that the first summand in (2.18) has the order O(ltl 112x x exp { -~ltl2p-l}) in the domain D_.

To prove an analogous estimate for the second summand in (2.18), we represent the function g( (, t) as follows:

(2.20) g((,t)=g-((,t)+g+((,t)= j ... dryds+ J ... dryds.

(Do\D+)uD_ D+

a) Let (71, s) E (Do\ D+) U D_. Then we have

j IF(y, s)l dy :( M L lsl'l7!lk [ 8( -ry) exp (- 41:l) + 8(71)] -00 k,>.

Here and below we denote by :L a finite sum over non-negative integer values k,>.

of k and real .X. By M we denote some positive number. Formulae (2.10), (2.12) imply the estimate

(2.21) l'ff(ry,s)l :( lsi-1/2A-1 Cls11/2) :( Mlsl-1/2exp (41:1)

Using the last two inequalities, we get

lg-((, t)l :( M L l(lkWiln ltllm, ((, t) ED_. k,>.,m

b) Now let (71, s) ED+. In this case the following inequalities hold:

lp(ry,s)l :( M G + 1:

1), I>T!(ry,s)l :( Mlryl- 1

,

~

J IF(y, s)l dy :( M L lsl'lln lsllml7!lklln l7!lln. _

00 >..,m,k,n

Using these inequalities, we get

lg+((,t)l :( MLWI!nltllm, ((,t) ED_. >.,m


Combining the obtained estimates, we conclude that the last summand in (2.18) is of the order 0 (lti"' exp { -~1ti2P- 1 }) in the domain D_. This completes the proof. D

The statement of Theorem 2.2 in terms of the variables 1J and t shows that the asymptotics of v2 ((, t) in the domain Do should be sought in the form of the series

1 00

V2((, t) =LIn' It I L IW2+s+m/2 R2,s,m(1J). s=O m=O

Substituting this series in (2.3), we obtain the recurrence system of ordinary differential equations

(2.22) R~,o,o- 2 (rJ + Zo(rJ))R;,o,o- 2 (Z~(rJ) + 4)R2,o,o = 0,

(2.23) R~,o, 1 - 2 (rJ + Zo(1J))R;,o,1 - 2 (Z~(rJ) + 3)R2,0,1 = 2 (Rt,o,1R2,o,o)',

(2.24) R~,1 ,0 - 2 (1J + Zo(1J))R;,1,0 - 2 (Z~(rJ) + 2)R2,1,o = 0,

~.o,m- 2 (rJ + Zo(rJ))R;,O,m- 2 (Z~(rJ) + 4- m)R2,0,m = -4R2,1,m-2 +

(2.25)

(2.26)

~,1,m- 2 (rJ + Zo(1J))R;,1,m- 2 (Zh(rJ) + 2- m)R2,1,m = m-1

= 2 L(R1,o,m-pR2,1,p)'. p=O

Solutions of equations (2.22), (2.23) and (2.24) can be written in a simple form

(2.27) R (rJ) = -b [41J2 2 81J3

1\.(rJ) _ 8(192

+ 1) /\.2(1J)] 2,0,0 + + 3ft 31f '

(2.28) R2,o,1(1J) = ~ [(-161J4 + 12)/\.(rJ)+

+ (481}; 401J) /\.2(1J)- 32(1}: + 1) /\.3(1J)] '


(2.29)

Such choice of solutions is due to the matching of asymptotics of the function v2((, t) in the intersection of D_ and D0 .

By direct calculations it is easy to show that the right-hand side of equation (2.25) for m = 2 is of the order 0( 196e-192

) as 19 -+ -oo. This implies that equation (2.25) form = 2 has a unique solution, which satisfies the asymptotic relation

(2.30) R2,o,2(11) = -2cp111 (0)192 + 2b + 0 ( 19"e-112), 19-+ -oo.

THEOREM 2.3. As t-+ -0, the asymptotic representation holds:

1 N

(2.31) v2((, t) = 2)n' ltl L IW2+s+m/2R2,s,m(19) + 0 (ltn, ((, t) E Do, s=O m=O

where N ) 0, R2,s,m E C 00 (lR), and v is some number, which tends to infinity as N-+ oo.

Proof. Consider the function 2

vg((, t) = L IW2+m/2 R2,o,m(19) + IW1 ln 1t1R2,l,0(19) · m=O

Then it follows from (2.3) that the difference r((, t) = v2 ((, t) - vq((, t) is the solution of the following initial value problem:

Cr = f, r((, 1) = R((),

~liv3 2 (£) where f = 6 11( - Cvq, R(() = 2:: R2,o,m 2 . Write out the solution m=O

where

8 { h((,t)} 8 { g((,t)} r((, t) = 8( w((, t) + 8( w((, t) '

00 ~

h((, t) = j G(ry- (, 1+ t) w(ry, -1) j R(z) dzdry = -oo -oo

00

{ 2 ~ = j G(ry- (, 1 + t) exp 8( -ry): } j R(z) dzdry =

-oo -oo 00

= j G(ry-(, l+t)ho(rJ)dry.

-oo


Formulae (2.27)-(2.30) imply that R(() = 0 ( (~'e-('/4 ), as (--+ -oo. There

fore ho('l]) is a function of polynomial growth, whence it follows that the last integral with all its partial derivatives converges uniformly. Taking this into account, we expand the function h( (, t) in the Taylor series

N

h((, t) = L hm,p(mtp + 0 (lti(N+l)o) = L ltlm/2 Pm('i9) + 0 (lti{N+l)o) · JmJ+JpJ(N m=O

The estimate of the remainder is valid in the domain D0 .

The function g( (, t) is defined by the formula

too 11 t

g((,t)= j j G(1)-(,t-s)1J!(1),s) j f(z,s)dzd1)ds= j C((,t,s)ds -1 -oo -oo -1

~

Denote by .\('1], s) the function 1J!(1), s) J f(z, s) dz. Then in the domain Do -00

we have

(2.32)

where

00

(2.33) () J am>.(2yFs,s) ( 2)d Cm o s = "' exp -y y,

~ UTJm

-oo

(2.34) 00

( ) _ ~ _ I 1

-p+t;z j 1 am+l>.(2yFs, s) ( 2) Cm,p S - ~ Cm,p,l S y 01)m+l exp -y dy,

1=1 -00

p;?: 1.

Combining expressions (2.32), (2.33), (2.34) and applying Theorem 2.1, we get the desired statement. D

THEOREM 2.4. As t--+ -0, the asymptotic representation holds:

N

(2.35) v2((, t) = L tmS2,m(() + 0 (lti''N+fl) , ((, t) ED+, m=O

where N;?: 0, S2,m E C 00 (0, +oo), a> 0, (3 E JR.

Proof. The solution of problem (2.3) has the form

where

t +oo

J J 1 { (ry-()2

} 3 g(C,t)= 2J1r(t-s)exp 4(t-s) 'I!(ry,sh(ry,s)dryds.

-1 -oo

By direct calculations one can show that in the domain D+

The statement of the theorem follows from the a priori estimates of solutions of a parabolic equation. The required inequalities are proved below.

Since t < 0, we have

t +oo

[g((,t)[ ( M j j ~exp{ (ry-()2} 3 4

[s[ iJ!(ry,s)[v1(ry,s)[dryds. -1-00

It is convenient to represent the integral in the right-hand side of this inequality in the form of the sum of three functions

t -isiP t isiP t+oo

g-((;, t) = j I .. dryds, l((, t) = j I .. dryds, g+((, t) = j J .. dryds . . -1-oo -1-lsiP -11siP

Estimate the function g-((, t). From Theorem 2.1 and estimate (2.21) it follows that

t -isiP _ j [s[-7/2 j 3 { (TJ _ ()2 r)2 } [g ((, t)[ ( M .;r=B [TJ[ exp 4[s[ + 4j8f dryds:::;

-1 -00


By definition, put f(w) = ZJ(w)A - 1(w). Then for any w E lR the inequality holds

f(w) ,; (1 + lwl3) exp(w2).

Combining Theorem 2.1 and estimate (2.21), we get

t lsi'

Jg0((, t)l ,; M j;;; j exp { (ry- ()2} ( 'T/ ) 4Jsl f 2Fs dryds ,;

-1 -lsiP

t

f I 14p-7/2 { (2 ( 1 } ,; M Fs exp -41sl-l + 21slp-l + 41sl2p-1 ds,; M[(].

-1

Finally, estimate the function g+((, t). t 00

Jg+((, t)l ( M j l=s j ry-4 exp {- (ry 41sf)

2

} dryds ( MC3+/2P

-1 isiP

This completes the proof. 0

THEOREM 2.5. For any k;:;, 3, the asymptotic representations hold as t-+ -0:

(2.36) +0 (lw exp { -~wp-l}) ,

[k/2] N

(2.37) vk( (, t) = I:; ln' Jtl I:; IW3k/

2+s+!+m/2 Rk,s,m( iJ) + 0 (ltJM), s=O m=O

N

(2.38) vk((, t) = I:; tm Sk,m(() + 0 (JtJ"), m=O

in the domains D_, D0 and D+, respectively. Here Fl(t) are some polynomials of degree l, a> 0, N;:;, 0, Rk,s,m E C""(JR), Sk,m E C""(O,+oo), and p,, v are some numbers, which tend to infinity as N -+ oo.

This theorem can be proved by induction with the help of methods used in proofs of Theorems 2.2, 2.3 and 2.4.

;;:


3. Asymptotics of the Solution Near the Origin. An expansion of the solution in a neighbourhood of the origin is sought in the form

00 [p/2]-1

(3.1) W= L"p/G L ln'cwp,s(~,T), p=2 s=O

where ~ = £-213x and T = £- 113t are new inner variables. Substituting this series into equation ( 1.1), we get the recurrence system of equations

(3.2)

(3.3)

owp,s {} (fwp,s) a;:-+ {j~

For brevity, we use the notation r(~, T) = w2 ,0 (~, T). Passing in series (2.1) from the variables (, t to the variables C T and equating the coefficients of £ 113 , we conclude that equation (3.2) must be supplemented by the condition

(3.4) f(~,T)=fiTI 1-3k/2Rk,O,O( }r), T-t-oo, (~,T)EDo, k=1 2 T

where D0 = {(~,T): T < -1~1 118 }, and the functions Rk,o,o are the same as in Theorems 2.1, 2.3, 2.5. Omitting the considerations leading to the explicit form of r(~, T), we immediately write out the formula

(3.5)

where

00

(3.6) (~, T) = j exp ( -~bs3 + TB2

- ~s) ds. 0

LEMMA 3.1. As T -+ -oo, the asymptotic representation holds:

00

(3.7) r(~, T) =I:: ITI-(3p+l)/2 Zp(11), (~, T) E DO', p=O

where Zp E 0 00 (JR'.); furthermore,

(3.8) 2

Zo('B) = -2?.9 + y7iA( t9),

(3.9)

Proof. The change of variables s = (e- 7)-112z in formula (3.6) yields

00 { } 1 4b 7 ~

<P 7 = ex - z3 + z2 - z dz. (~, ) Je _ 7 f P 3J(~2- 7)3 e- 7 Je- 7

0

Expanding the integral in a Taylor series, we are led to the asymptotic representation

tuting the last expansion into (3.5), we have

(3.10) r(~, 7) =-~ 171-(3p+l)/

2 ( ~ r t; Em(t9)Fp-m(t9),

where Fn(t9) = ~ ( H3n+J(t9) }nS3n(t9)A(t9)) , Eo(t9) = 1,

Em(t9) = f(-1)k L i/.ik! (.Hai,(t9)- ~s3i,-J(t9)A(t9)) x ... k=l t1 + ... +tk=m

whence one can get expressions (3.8), (3.9) by direct calculations. D

LEMMA 3.2. The functions Zp(t9) are of polynomial growth as t9 -.> +oo. The estimate Zp(t9) = o(t93P- 1 ) holds as t9-.> -oo.

Proof. From (3.10), we see that

whence the first assertion of the lemma follows, since A( 19) = 0 ( 19) as 19 -+ +oo, and Hk(19), Sk(19) are polynomials.

By formulae for Em(19) and Fm(19), we get

- - (~)p { (219)3P+l p (219)3(p-m)+l m - k (219)3m Zv(19) - 6 p! + L (p- m)! L( 1). L it!. .. ik! +

m=l k=l t1 + ... +tk=m

=- (~)P { (219)3p+l + (219)3p+l...;.... 1 ~ (-1)k X 6 p! L..... (p - m)! L..... m!

m=l k=l

X~( -1)' q (k- s)m + ~ ~ bp,m,k193p-2k-1 + 0(193p-3)} = 0(193p-3).

This proves the second assertion of the lemma. D

THEOREM 3.1. The function r(~, r) defined by formulae (3.5), (3.6) is a solution of problem (3.2), (3.4).

Proof. Clearly, the function <!?(~, r) satisfies the heat equation. Therefore the function r(~, r) satisfies equation (3.2). By virtue of lemma 3.1, it remains to prove that Zp(19) = Rp+I,o,o(19) for any p ~ 2.

Equation (3.2) implies the recurrence system of ordinary differential equations for Zp ( 19):

(3.12) Z~-(ZJ+219Z0)'=0,

(3.13) z~- 2 (19 + Zo)z;- 2 (Zb + 4)Zt = o,

p-1

(3.14) z;- 2 (19 + Zo)z;- 2 (Zb + 3p+ 1)Zp = 2 L Zp-mz:n, p ~ 2. m=l

For p ~ 2, the homogeneous equation (3.14) has two linearly independent solutions, one of which is of the order 0( {)3P-l) as {) -+ -oo, while the other one is of the order O(e112

) as{)-+ +oo. From this fact and lemma 3.2 it follows that, for p ~ 2, the functions Zp( 1J) are uniquely determined by equations (3.14). By construction, the functions Rk,o,0 (1J) satisfy equations (3.14) with p = k - 1. Furthermore, Theorem 2.5 implies that these functions have the same order as Zp(1J) at ±oo. Hence it follows that Zp(1J) = Rp+1,0,0(1J). D

The obtained results make it possible to construct an asymptotic expansion of the solution of problem (1.1), (1.2). This expansion has different forms in different subdomains of the strip S. The most significant corollary is the following.

THEOREM 3.2. In the domain {(x, t) : lxl < c:", -c:f3 < t < 0}, 0 < a < 2/3, 0 < (J < 1/3 the asymptotic representation holds as c: -+ 0:

u(x, t; c)= c: 113 r(~, 7) + a(c:113),

where the function r(~, 7) is defined by formulae (3.5), (3.6).

REFERENCES

[1] A.M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, Amer. Math. Soc., Providence, 1992.

[2] V.G. Sushko, On asymptotic expansions of solutions of a parabolic equation with a small parameter, Differentsial'nye Uravneniya, 21, 10 (1985), 1794-1798.

[3] E. Hopf, The partial differential equation u, + uu, = !JUxx. Comm. Pure Appl. Math., 3, 3 (1950), 201-230.

VOLUME 8 2001, NO 3-4 PP. 273-285

DENSITY ESTIMATE IN SMALL TIME FOR JUMP PROCESSES WITH SINGULAR LEVY MEASURES AND ITS

SUPPORT PROPERTY *

Y. ISHIKAWA t

Abstract. In this paper we study the asymptotic behaviour of the transition density for processes of jump type as the time parameter t tends to 0. We are interested in the case, where the support of the Levy measure of the driving process is very singular. The main result is that, under certain restrictions, the density behaves in polynomial order or may decrease in exponential order as t --t 0 according to geometrical conditions of the objective points. We further study the support property of so-called canonical jump processes.

Introduction. In what follows we study two types of Markov processes x,(x) of jump type on Rd. Analytically, we can associate to each Markov process a semigroup (T,)t2:0 (on C8"(Rd)) in such a way that T, = p,(x,dy), the right-hand side denoting the transition function of x,(x).

In terms of potential theory, the infinitesimal generator A of the semigroup (Tt) is given in general form by A= P + S, where

Pf(x) =< gradf(x),b(x) >

and

Sf(x) = j[f(x +y)- f(x) < gradf(x),y > ·l{IYI:SIJ]K(x,dy).

Here K is a non-negative kernel such that J (lyl 2 11 1 )K(x, dy) < +oo. In some case, to the semigroup (T,)t:2:0, T, = e'A, there corresponds a Markov process

• The author sincerely thanks Professor A. L. Skubachevskii and all the members of the organizing committee for the hospitality at MAL

t Department of Mathematics, Ehime University, Matsuyama 079 Japan

273

274 Y. ISHIKAWA

x1 ( x) given by stochastic differential equation ( 1.1) below. The process t >-t x1(x) is a jump type process in case S # 0, that is, in case A is an integrodifferential operator. The kernel K and the measure tt which characterize x1 ( x) are related by

j K(x, dy) = j 1A\{o)('Y(x, ())tt(d() A

for a Borel set A C Rd. Here the measure tt is called the Levy measure associated to A. Hence we start conversely from the process { Xt ( x)}, q,nd study the short time properties of T1 = p1(x, dy) as the transition function of Xt(x).

Analysis on jump type processes driven by Levy processes have some history. Processes driven by the Poisson process were studied by CarlenPardoux [4], and Elliot-Tsoi [5]. Special types of those processes were analysed by Bichteler-Jacod [3] and Tsuchiya [24]. After these, a theory to guarantee the existence of the transition density has been developed by Picard [19], [20], [21].

In Part I below we study the case when the coefficient of x1(x) with respect to the driving jump is non-degenerate. In Part II we study so-called canonical processes instead of standard jump processes when the coefficients (viewed as vector fields) may degenerate. In both cases we show a precise short time asymptotic bounds for the transition density p1(x, y), the density of p1(x, dy) with respect to d-dimensional Lebesgue measure. These calculations are carried out in terms of stochastic analysis of variations (Malliavin calculus).

Part I

1. Introduction to Part I. Consider the semigroup (Tt)t2:0 on C0 (Rd) corresponding to the jump process given by

t c

(1.1) Xt(x) =X+ J b(x,(x))ds + L r(x,(x), t.z(s)), Xo(x) - x. 0 sS:t

c Here b(-) : Rd -+ Rd, r(·, ·) : Rd X Rd -+ Rd, L denotes the "compensated

sSt sum" ([2]), and z(s) denotes ad-dimensional Levy process whose Levy mea-

JUMP PROCESSES WITH SINGULAR LEVY MEASURES 275

sure is given by f-!(d(). That is, (1.2)

"¢1(0 = E[eie·z(t)] = exp (it.;· c + t j (eie·< - 1 i.; · (1{1(1SIJ)f-!(d()) .

Or equivalently,

(1.3)

t t

x1(x) = x + j b'(x,(x))ds + j j 'Y(x,(x), ()N(dsd()+

0 0 1<19 t

+ j j 'Y(x,(x), ()N(dsd(),

0 1<1>1

where N denotes the compensated Poisson random measure : N(dsd() = N(dsd()- dsf-!(d(), and b'(x) = b(x)- J "f(X, 0!-!(d().

1(12:1 It is known that, under certain conditions on f-! and 'Y (see below), the

SDE has a unique solution, and the map x ,.__; x1(x) is a stochastic flow of diffeomorphism. We denote it by x 1(x) = ¢;,,1(x,(x)), s < t. We denote by Pt(x, dy) the transition function corresponding to this semigroup. Under some assumptions the transition function p1(x, dy) possesses a density p1(x, y), which belongs to C'{' : p1(x, dy) = p1(x, y)dy. This is due to the so-called calculus of Malliavin of jump type ([11], [9], [19]). More precisely,

THEOREM I.l ( CF. [21]). Under assumptions (A.1) - (A.4) stated below, the density Pt ( x, y) exists and satisfies the following estimate:

d

(a) SUPx,yPt(X, y)::; CoC~ as t-+ 0.

(1.4) d

Pt(x,x) X C~ as t-+ 0 uniformly in x.

(b) for all k E Nd, there exiBts Ck > 0 such that

(1.5) (k) (lk)+d)

supfpt (x,y)[::;Ckr ~ as t-+0. x,y

Here p(k) denotes the k-th derivative of p with respect toy.

Hence, throughout this paper, we suppose that there hold assumptions (A.l)-(A.4) stated below. We shall further study these bounds with respect to x, y. Namely, given x, y, we study the asymptotic behaviour of p1(x, y) as t-+ 0.

276 Y. ISHIKAWA

We define a series of functions (An);:'=0, An : Rdx(n+l) -t Rd as follows: Ao(x) = x,

We fix x E Rd, define the map \1? : (xb ... , Xn) >--+ An(x, x1, ... , Xn), and put Sn to be the support of the image measure of J.l®n by \1?, S = USn·

n

DEFINITION (accessible points). Points in S are called accessible points. Points in S \ S are called asymptotically accessible points. Points in Rd \ S are called inaccessible points.

We put the set Px as Px = x + {'y(x,();( E suppv}. Further put p~n) = {y E PZn-liZ1 E Px,Zi E Pz,_pi = 2, ... ,n-l}, n = 1,2, ... (zo = x). Then P~1 ) = Px, and the set P~n) may be interpreted as those points, which can be achieved from x by n jumps of Xt(x). We let for x, y E Rd(y # x)

" a(x, y) = the minimum number l such that y E P~1) if such l exists, " a(x, y) = +oo if not.

Or equivalently, a(x,y) = inf{n;y E U Sk}. ~ k~n

2. Assumptions. Here we sum up our assumptions. We assume the following (A.l)-(A.4).

(A.l) The Levy measure is such that there exists some 0 < (3 < 2 and positive cl, c2 such that, as p -t 0,

(1.6) C~l-fJ I~ J ((*J.!(d() ~ C2p2-fJ I

i(i:SP

The inequalities should be interpreted in the sense of those of matrices. Or, equivalently, as p -t 0, for all u E sd-I, J < (, u >2 J.l(d() X p2-fJ. In

i(i:SP particular,

(3 = inf {a; J [(["J.l(d() < +oo} , i(I:Sl

hence this coincides, under (A.l), with the Blumental-Getoor index (3 for the process z(t).

(A.2) We further put the following assumptions with respect to the (3 in (A.l):

(2-a) If 0 < (3 < 1, we assume c = J (t.t(d(), b = 0 and, asp-+ 0, for 1(1:9

(1.7) J < (,u >2 1{<(,u»Oj(()t.t(d() ~ l-!3. {1(19}

(2-b) If (3 = 1, then lim sup J (t.t(d() < +oo. <->0 {e<l(i$1}

(A.3) (3-a) For any p 2 2 and any k ENd\ {0},

(3-b) There exists 6 > 0 such that

(A.4) We assume, for some C > 0,

(1.9) }nf ldet (I+ ~/' (x, o) I > c. xER ,(Esuppp, X

The conditions (A.3) and (A.4) guarantee the existence of the flow ¢,1(x)(w) : Rd -+ Rd,x H> x1(x) of diffeomorphisms for all 0 < s :S t. We remark a crucial condition for the existence of the density in [19]: for each t > O,p 2 1,

with '1/;, = (;x¢,,1(x,(x))) (~~(x,(x),o)) is derived from (1.8) and (1.9)

by an argument similar to what stated in the proof of corollary 4.4 of [19].

278 Y. ISHIKAWA

3. Result in part I. As a concrete example we consider the following case. We suppose that the process Xt(x) satisfies the assumptions (A.1)(A.4). Assume the Levy measure It of z(s) to be given by

00

(1.10) jt(d() = L kn&an (d(). n=O

Here (an;n E NU{O}) denotes a series of points in Rd, and (kn;n E NU{O}) denotes a series of real numbers satisfying ian I decreases to 0 as n -+ oo,

00

kn > 0, and L; kniani 2 < +oo. We further assume n=O

N = N(t)- max{n; iani > tl/13} ~ log G)· Then we have the following theorem.

THEOREM I.2 [13] {upper bound for asymptotically accessible points).

(a) Assume that yES (a(x, y) < +oo). Then

Pt(x, y) X t'"(x,y)-d/fJ as t-+ 0.

{b) Assume y E S\S (a(x,y) = +oo), b(x) := 0, and fix (3' > (3. Then as t-+ 0, logpt(X, y) is bounded from above by r = r(t) : (1.11)

r - -min~ ( Wn log ( t~n) + log( wn!)) + 0 (log ( ~) log log ( ~)) . Here the minimum is taken over all choice of a0 , ... , aN by l;n for n =

1, 2, ... , n 1 and n 1 E N, which satisfy

(1.12)

For the lower bound, as t-+ 0, logpt(x, y) is bounded below by

for some c > 0.

REMARKS. (a) Conditions (1.12) and N(t) x log(t) are complimentary. That is, condition (1.12) becomes more strict when t -+ 0. However, one can use more (an)f:~o in finding the minimum by the condition on N.

(b) In (1.11) N x Jog(}), hence ithe first termi 2: C(Jog(t)}2. This

implies r < 0 if t > 0 is very small. (c) Assume ?(X,() = (, and Jet x be rational (say, x is the origin) and

(an) binary (e.g. an= ((-2)-n, (-2)-n) in cased= 2). Then for almost all yES= [-1, 1] x [-1, 1] with respect to the Lebesgue measure we have the case {b) above.

Part (a) of this theorem follows similarly to the arguments in Picard [21]. The former statement of (b) is an extension of a result in [20], which treats the case of 1-dimensional Levy processes. The latter half of (b) is new. For its proof, we construct concrete trajectories s >-7 x8 (x) for which the end point Xt(x) is quite near toy. We show that such trajectories have positive probability, and calculate it. Determining the constant c > 0 is not easy. Also, to determine the set S in R d is a delicate problem, and is left in the future.

Proof (a). We prove the following two lemmas (corresponding respectively to the lower and the upper bounds), and apply them with ?n = 0 if a(x, y) :::; n, = +oo if a(x, y) > n, Kn > 0 small and 'Pn = id. D

LEMMA I.3 {lower bound for accessible points). ' Let (~n)nEN be an Rd-valued random variables {i.i.d.) obeying the probability law v(d(), independent on z(s). We put a Markov chain (Un)nEN by Uo = x, Un+l = Un + ?(Un, ~n+l), n E N. Assume that for y E Rd, there exists some n 2: 1, 1' = ?n 2: 0 and c > 0 such that for all c E (0, 1] P(IUn Yi :::; c) 2: cc~. Then we have

Pt(X, y) 2: Ctn+(~-d)/(3 as t-+ 0.

Let ('Pn)nEN be a series of smooth functions: Rd-+ Rd. We define another Markov chain (Vn)nEN by Vo = 'Po(x), Vn+l = Vn+('Pn+lo?)(Vn,~n+l)· Further we put the series of real numbers ( <Pn)nEN by

<Pn = sup (I'Pk(Y)- Yl + I'P~(y)- II). k<n,yERd

After these preparations we have

LEMMA I.4 (upper bound). Choose y f x. Assume that there exist a sequence bn)nEN, ?n E [0, oo], and a non-decreasing sequence (Kn)nEN, Kn > 0 satisfying the following condition: for each n, for any ( 'Pk)~=o satisfying <Pn :::; Kn, Vn defined as above, there exist some Cn > 0 such that

if ?n < +oo then P(IVn- Yl:::; c) :::; Cnc~" for all c > 0,

280 Y. ISHIKAWA

and

if "'n = +oo then P(!Vn- yJ :::; c)= 0 for c > 0 small.

Further we put r = minn(n + bn- d)/ /3). Then we have; Iff< +oo then Pt(x, y) = O(tr) as t-+ 0. If r = +oo then for any n EN p1(x, y) = o(tn) as t-+ 0. The detailed proof of Theorem 1.2 (b) is given in [13], and we omit it.

Part II

4. Introduction to Part II. In this part, we denote by x1(x) another process, which is Rd-valued and of jump type, given by the SDE

m

(2.1) dxt(x) = L XJ(x 1_(x)) o dzJ(t), xo(x) = x j=l

where z(t) = (z1(t), ... ,zm(t)) denotes an Rm-valued Levy process (martin-gale), and X 1(x), ... , Xm(x) are Rd-valued functions on Rd (viewed as the vector fields), and odzJ(t) means the canonical integral. That is,

m

dx1(x) = LXJ(Xt-(x))dzJ(t)+ j=l

+ { Exp (t~zJ(t)XJ) (x1_(x))- x1_(x)- t~zJ(t)XJ(Xt_(x))}.

Here the second term is a sum of terms of order (~z(s)) 2 , and 7/J(t,x) _ Exp(tv)(x) is the solution flow of the differential equation

d7j; dt(t,x) = v(7j;(t,x)), 7/J(O,x) = x.

In [18] this process is called as "Stratonovich" stochastic integral. We minic their notation odz(t) in this paper. SDE (1.4) has a unique solution which is a semimartingale [18] Th. 3.2). We denote by Pt(x, dy) the transition function of x1(x).

In this note we study the existence, support property and the short time behavior of the transition density p1(x, y) attached to x1(x), when the support of the Levy measure of z(t) is very singular (i.e., it is countably supported).

Let z(t) be a Levy process, Rm-valued, with Levy measure f.t(d(): J (1(12 A l)f.t(d() < +oo. That is, the characteristic function 1/Jt is given

R"' by

t

We may write z(t) = J J ((N(dsd() -1{1<1::;1) • f.t(d()ds), where N(dsd() 0 R"'\{O)

is a Poisson random measure with mean >.-(dsd() = ds x f.t(d(). For the clarity of notation, we may write N = >.+, >. = >.+- >.-, and z(t) = (z1 (t), ... , Zm(t)).

We define an m x m matrix V(p) and a positive v(p) by

(2.2) V(p) = J (1(f.t(d(),

{l<l:;p)

v(p) = j ICI 2f.t(d()

{I<I:SP)

respectively. We put

B - 1· . f V(p) = 1mm -( -)

p-+0 v p

as a component-wise limit. Consider in particular the Levy measure

m m

(2.3) f.t = I>j = L. Tj* f.tj j=1 j=1

Here T1* fli - flJ o r1-1

, f.tJ is a 1-dimensional Levy measure of the form 00

I; kn5{±iin)0 and Tj : (j H (0, ... , 0, (j, 0, ... , 0). We let n=l

00

n=O

where

kn = 2nf3,(J E (0,2), and iin = 2-n.

In this case, for p > 0 small,

v(p) = J l(l 2f.t(d() = f L 2-n(Z-!3) X p2-!3.

1(19 J=l n,z-nsp

282 Y. ISHIKAWA

This is equivalent to p( {1(1 > p}) x p-f3 as p -> 0 in our setting. Further, we choose points (ak)kENu{o} in Rm defined by

m

ak = L l{[~J-j+!EmZ)(k)Tj(( -l)kiil2!;),

j=l

and

(2.5)

Then we have N(t) x logO). Here we denote by [x] the integer part of x. We carry out our study in the probability space (0, (.Ft)o::;t:s;T, P), where

0 = D[[O, T], Rm] (the Skorohod space), (.Ft)o::;t:s;T- filtration generated by z(t), P probability measure on 0 of z(t). That is, p(d() = P(z(s + ds) E

d(lz(s)l/ds. Let 0 1 be the set of all integer valued measures on [0, T] x Rm, and .F1

be the associated a field. This measure can be realized as Stieltjes measure associated with integer valued cadlag paths wE 0 = D[[O, T], Rm], which is of bounded variation. Let W be the collection of Rd-valued, .F1 measurable random variables F(wl),w1 E 01.

In the sequel we follow Picard [19] and Privault [23] for the formulation of the analysis on the Poisson space. For each u = ( t, () E [0, T] x Rm, operations c~, c;;-, both from nl to nl> are given by

c;;-w1(A) = w1(An { u }c), E~w1 (A) = c;;-w1(A) + lA(u), A E B([O, T] x Rm).

A difference operator Du on W is defined following Picard [19] by

DuF = F o c~ - F.

ForT= (ub ... , Ut), we put c; = €~1 o ... oc~1 , D7 = Du, ... Du,· We remark that Du does not satisfy the property of the derivation, since

Du(F1F2) = F1DuF2 + F2DuF1 + DuF1DuF2.

We denote by oi the partial derivative with respect to (i. For a multiindex a = ( a 1 , . .. , am), we put 8"' = o"'' ... fJ"'m. We will define the partial derivative of ( >--+ D(t,()(F). Note that, by definition, the meaning of this notion differs from the notion of "derivative" of F in the usual sense. We denote by fJD(F) = fJD(t,()(F) the d x m matrix

(2.6) fJD(F) = (81Dt,(F ... 8mDt,<F)

of "second" differentials of F E W whenever these derivatives exist. This notion will play an important role in the analysis on Poisson space cf. [22].

5. Results in Part II. We put the following assumptions (B.I), (B.II), (B. III).

(B.I) For some a E (0, 2),

0 < liminfv(p) < limsup v(p) < +oo. p-+0 p" - p->0 p"

(B.II) For some c > 0,

(2.8) B ~ cl.

The Levy measure tJ- given by (2.3), (2.4) satisfies (B.I) above with a= 2-/3, and, (B.II) with c = 1.

Assume that £ denotes the Lie algebra generated by X 1 , ... , Xm, and .C(x) denotes the space of tangent vectors {Xx; X E £} at each x E Rd. Here Xx denotes the restriction of X at x. We put the following assumption, referred to as restricted H ormander condition.

(B.III) For all x E Rd .C(x) ""Rd. We hope to approximate the trajectory (and the end point) oft>-+ x 1(x)

by a Markov chain. To this end, we introduce the following. Let (Cn)nEN' Cn : Rdxn -+ Rd be a deterministic chain of functions

given by

Co(x) = x,

Let ('Pn)nEN' 'Pn : Rd -+ Rd be smooth functions. We introduce the norm (<Pn)nEN by

<Pn = sup (I'Pk(Y) - Yl + I'P~(y) -II). k::;n~yERd

n

As in Part I, we call points in S accessible points in canonical sense, and points in S \ S asymptotically accessible points. We put for x, y E R d,

(2.10)

284 Y. ISHIKAWA

Via similar arguments as in Kunita (17], (14], we can prove the following theorem.

THEOREM II.l ( CF. (17], (14]). (a) Assume supp J.t is compact. Under (B.I), (B.II) and (B.III), the density of the law of Xt(x) exists for all x E Rd and all t > 0: Pt(x, dy) = Pt(X, y)dy.

(b) Further, in case J.t is given by (2.3), (2.4), we have for all x E Rd and all t > 0 supppt(x, .) =Rd.

Further we have the following short time asymptotic bounds of Pt(x, y) when the Levy measure is given by (2.3), (2.4).

THEOREM II.2 (14]. Let J.l is given by (2.3), (2.4). Suppose that (B.III) is satisfied. Let y =I x. Then we have the following estimates for the density Pt(x,y).

(a) Assume that yES, that is, a(x, y) < +oo. Then we have

Pt(X, y) :S C t"'(x,y)-d/fJ as t-+ 0.

(b) Assume y E S\S (a(x, y) = +oo). Choose (J' > (J. Then !ogpt(x, y) is bounded from above by the expression of type r = r(t):

as t -+ 0. Here the minimum is taken with respect to all choice of a0 , ... , aN by Xn for n = 1, 2, ... , n 1 and n 1 E N such that

(2.11)

N

where Wn = # of an in the choice and n 1 = L.; Wn· n:::::O

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[3] K. Bichteler and J. Jacod, Calcul de Malliavin pour les diffusions avec sauts: existence d'une densite dans le cas unidimensionel, Seminaire de Probabilites XVII, Lecture Notes in Math., 986, Springer-Verlag, Berlin (1983), 132-157.

[4] E. Carlen and E. Pardoux, Differential calculus and integration by parts on Poisson space, Math. Appl., 59 (1990), 63-73.


[5] R. Elliot and A. Tsoi, Integration by parts for Poisson processes, J. Multivariate Anal., 44 {1993), 179-190.

[6] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference an Harmonic Analysis {1981, Chicago), Wadsworth, Belmont, {1983), 590-606.

[7] T. Fujiwara and H. Kunita, Stochastic differential equations for jump type and Levy processes in diffeomorhisms group, J. Math. Kyoto Univ., 25 (1985), 71-106.

[8] T. Fujiwara and H. Kunita, Canonical SDE's based an semimartingales with spatial parameters; Part I Stochastic flaws of diffeamarphisms, Part II Inverse flaws and backwards SDE's, Preprint {1999). .

[9] Y. Ishikawa, Asymptotic behavior of the transition density for jump-type processes in small time, Tahaku Math. J., 46 {1994), 443-456.

[10] Y. Ishikawa, On the upper bound of the density for truncated stable processes in small time, Potential Analysis, 6 {1997), 11-37.

[11] Y. Ishikawa, On the lower bound of the density for jump processes in small time, Bull. Sc. Math., 117 (1993), 463-483.

[12] Y. Ishikawa, Large deviation estimate of transition densities for jump processes, Ann. l.H.P. Prababilites, 33 {1997), 179-222.

[13] Y. Ishikawa, Density estimate in small time for jump processes with singular Levy measures, submitted, (1998).

[14] Y. Ishikawa, Existence of the density far a jump process and its support, Preprint, {1999).

[15] H. Kunita, Stochastic differential equations with jumps and stochastic flows of diffeomorphisms, Ito's stochastic calculus and probability theory, Springer-Verlag, Tokyo, {1996), 197-211.

[16] H. Kunita, Canonical SDE's based on Levy processes and their supports, Stochastic Dynamics, Springer-Verlag, Berlin, {1999), 283-304.

[17] H. Kunita and Oh, Existence of density of canonical stochastic differential equation with jumps, Preprint, (1999).

[18] T. G. Kurtz, E. Pardoux and P. Protter, Stratonovich stochastic differential equations driven by general semimartingales, Ann. Inst. Henri Poincare, 31 {1995), 351-377.

[19] J.Picard, On the existence of smooth densities for jump processes, Probab. Th. Relat. Fields, 105 {1996), 481-511.

[20] J.Picard, Density in small time for Levy processes, ESAIM Probab. Statist., 1 {1997), 358-389 {electronic).

[21] J.Picard, Density in small time at accessible points for jump processes, Stochastic Processes and their Applications, 67 {1997), 251--279.

[22] N. Privault, Chaotic and variational calculus in discrete and continuous time for the Poisson process, Stochastics and Stochastic Reports, 51 {1994), 83-109.

[23] N. Privault, A pointwise equivqlence of gradients on configuration spaces, C.R. Acad. sci. Paris, 327 (1998), 677-682.

[24] M. Tsuchiya, Levy measure with generalized polar decomposition and the associated SDE with jumps, Stochastics and Stochastic Reports, 38 {1992), 95-117.

VOLUME 8 2001, NO 3-4 PP. 287-295

MOV][NG BOUNDARY PROBLEMS FOR NONLOCAL FUNCTIONALS *

G.A. KAMENSKH t

Abstract. Moving boundary problems for nonlocal functionals are studied. Generalized necessary transversality conditions for these problems are deduced.

1. Introduction. Consider the problem of extremum of the functional

X!

(1) j F(x, y(x), y(a(x)), y'(x), y'(a(x))) dx -+ extr,

xo

where the deviation a(x) is of the retarded type: x ;::: a(x). Suppose that a(x) E Cl, is monotonous and that the inverse function 'Y = a-1 exists and 'Y E C1

.

The two following types of boundary value conditions are possible: I. A boundary function <p is defined on [a(x0), x0], and there are stated

the asymmetric boundary value conditions

(2) y(x) = cp(x), x E [a(xo),xoJ,

(3)

where y1 is a given constant.

* With support by the Russian Foundation for Basic Research (Grant N 01-01-00038) and by the St. Petersburg State University (Grant N E20-1-l.0-194).

t Department of Differential Equations, Moscow State Aviation Institute, Volokolamskoe shosse 4, Moscow 125871, Russia

287

288 G.A. KAMENSKII

II. Boundary functions <p and 1/J are defined on [a(xo), x0] and [a( xi), xi], correspondingly, and there are stated the symmetric boundary value conditions

(4) y(x) = <p(x), x E [a(xo), xo],

(5) y(x) = 1/J(x), x E [a( xi), x1].

L.E. El'sgol'z was the first to study asymmetric variational problems for nonlocal functionals ([1], Chap. 6). Later, the theory of symmetric variational problems for nonlocal functionals began to develop [2-5]. Asymmetric and symmetric boundary value conditions occur also in the theory of optimal control of systems with delay. The maximum principle for an asymmetric problem was formulated in [6-7]. Asymmetric variational problems, along with symmetric ones, are interesting both from theoretical viewpoint and in view of their applications. Specifically, a symmetric problem arises when it is required to investigate the behavior of a system for x > x 1 ; then the function 1/J can be taken to be the initial one. (See e.g.,[6]).

The example of such a problem is Krasovskii's problem of damping of a controllable system with delay [8, Chap. 11, §45]. In this book the following control problem is considered. Suppose that for the control system

y'(t) = Ay(t) + By(t- T) + Cu(t)

with initial condition

y(t) = <p(t) t E [to- Tj

it is necessary to find such a control function u(t) that fort 2: t 1 the solution y(t) = 0 and in addition

t1 +r J u2 (t) dt-+ min

to

This problem is equivalent to the symmetric variational problem for the functional

tt+r J [y'(t) - Ay(t)- By(t- Tw dt to

MOVING BOUNDARY PROBLEMS FOR NONLOCAL FUNCTIONALS 289

with the right boundary condition

More general problem of damping of a controllable system with delay was considered in [9], see also [10]. If, however, the behavior of the system for x > x1 is of no interest and it is only required to bring the system to the point Y1> then an asymmetric problem arises.

In some problems of cosmic navigation it is necessary to bring the object not to the fixed point of space, but to any point of a given trajectory. Then there arise the problem of extremum of a nonlocal functional with a moving boundary that is studied here.

The asymmetric problems were studied in [11] ( see also [12-14]) under following suppositions. The admissible function y belongs to the space AC[x0 , x1] of absolutely continuous functions on [x0 , x1], <p E AC[a(x0 ), x0].

The derivatives of these functions exist almost everywhere. It is also supposed that the derivatives y' belong to the space Lp[x0 , x1], 1 ::; p ::; oo. The function F is assumed to be twice continuously differentiable with respect to all arguments and to satisfy the growth constraints

for some continuous functions K > 0 and L > 0. It was proved in [8] that under these conditions integral (1) exists for any admissible function y.

Introduce the function F = 1'(x)F('Y(x), y('Y(x)), y(x), y'('Y(x)), y'(x)) and <li = F + F, and put

for

for

In [11] it was proved the following theorem.

290 G.A. KAMENSKII

THEOREM 1. Suppose that all above mentioned assumptions concern-ing functions F and a are fulfilled and that an extremum of functional (1) under boundary conditions (2), (3) is attained at y •. Then y. satisfies almost everywhere on [x0 , xJ] the equation

xr

(7) J Wy(x) dx- Wy'(x) =C.

xo

Here Fy(x) and Fy'(x) are partial derivatives ofF with respect to the second and to the fourth arguments, and Fy(x) and Fy'(x) are partial derivatives ofF with respect to the third and to the fifth arguments, correspondingly.

We can differentiate equation (7) with respect to x and write it in the form

(8)

Equation (8) is a generalized Euler equation for functional (1). The solution y satisfies (8) almost everywhere on [x0 , x!] and is a generalized solution of this equation. Note that the second derivative y" may or may not exist, but the composite function Wy'(x), as it follows from (7), is an absolutely continuous function and has almost everywhere on [x0 , x1] the derivative equal to Wy(x)· (See [8].)

2. Moving Boundary Variational Problem for Nonlocal Functional with One Deviation of Argument. Consider the problem of extremum of functional (1) with boundary condition (2), but now the point x1

is not fixed, and instead of condition (3), another condition is stated. It is given a function g(x), g E C 1 and

(9)

We derive here the necessary transversality condition for problem (1), (2), (9).

Suppose that y. is a solution of this problem and x1 is the abscissa of the point of intersection of graphs of y.(x) and g(x). The set of admissible functions in this problem includes the set of admissible functions in problem (1),(2),(3) with a fixed right-hand point (x1 , y1), y1 = y.(xi) and therefore must satisfy equation (8) almost everywhere on [x0 , x1].

We assume here that

(10)

Then we can consider a more simple space of admissible functions. It will be the space D of continuous functions having piecewise continuous derivatives. The set Q of possible discontinuities of derivatives has the following structure: it contains the points q1 = 'Y(xo), ... , qk = 'Y(qk-1), ... and the points ih = a( xi), ... , iik = a(iik-1), .... From supposition that a(x) < x it follows that Q is a finite set. If x E Q, then a(x) E Q and 'Y(x) E Q and y'(x- 0) and y'(x + 0) exist. If x1 changes, then Q changes but remains finite.

THEOREM 2. Suppose that all conditions of Theorem 1 are fulfilled, a(x) < x andy. is a solution of problem (1), (2), (9). Then y. satisfies the transversality condition

(11) [F + (g' - y')Fy'] lx=x1 = 0.

Proof. Let y.(x) be the function that gives the minimum value to functional (1) with conditions (2),(9), and (x1, y1) be the point of intersection of graphs y.(x) and g(x).

Denote by Yp,(x), (fl. > 0) an arbitrary continuously differentiable continuation of y.(x) on [x0 , x1 +fl.], and by y, we denote the restriction of y'"(x) on [xo, x1 + c] forcE [0, fl.).

Define

(12) w(c) = g(x1 +c)- y,(x1 +c), ry(x1 +c)

where ry(x) is a fixed function such that 1) E C1[x0 , x 1 +fl.], ry(x0 ) = 0, rl(x1) = 0, ry(x) f. 0 on [x 1 ,x1 + !J.], and B(x,c) = y.,(x) +w(c)ry(x).

It follows from (12) that for sufficiently small c > 0 the function B(x, c) belongs to the set of admissible functions of problem (1), (2), (9). From the supposition that functional (1) attains a minimum on y.(x) it follows that J(B(x, £)) > J(y.) for sufficiently small c > 0.

Define the function x1+e

r(c) = j F(x,B(x,c),B(a(x),c),B'(x,c),B'(a(x),c))dx. XQ

The function r(c) is differentiable and

r'(c) = F(x 1 +c,B(x1 +c,c),B(a(x1 +c),c),B'(x1 +c,c),B'(a(x1 +c), c))+

X1+e

+ .I w'(c)[Fyry(x) + Fy01)(a(x)) + Fy'r/(x) + Fv~1J'(a(x))] dx.

xo

292 G.A. KAMENSKII

It follows from J(O(x,c:)) > J(y.) that r'(O) = 0. xt+e x1+e

In the. integrals J Fya(x)7J(a(x))w'(c:) dx and J Fy~7J'(a(x))w'(c:) dx xo xo

we change variables putting a(x) = z,x = l'(z) and come back to denoting by x the integration variable. Then putting c: = 0 and using the definition of the function , we obtain

a(x,) x1

+w'(O)[ 1 [ii>y7J(x) + ii>y'7J'(x)] dx + 1 [Fy7J(x) + Fy'11'(x)] dx, xo a(xi)

or Xl

r'(O) =Fix, + w'(O) 1 ['l!y1J(x) + 'l!y'11'(x)] dx. xo

The function 'I! satisfies equation ( 8) and therefore

Xl X1

1['1!yr] + 'l!y'111] dx = 1 :X ('l!y'1J) dx = 'l!y'1Jix, - 'l!y'1Jixo = 'l!y'11lx1

xo xo

because 7J(x0 ) = 0. It follows now from (12) that

(13)

Evaluating w'(O), we have

and from (13) we obtain

r'(O) =Fix,+ 'l!y'(g'(xl)- y'(xl)).

From supposition that a(x1) < x1 it follows that 'l!y'lx, = Fy'lx,· D

REMARK. If condition (10) is weakened to

a(x) < x for x E [xo, x1)

and a(xl) = x1, then in Theorem 2 condition (ll) must be replaced by

ii> - (g' - y')ii>y' lx~x, ·

3. Moving Boundary Variational Problem for Nonlocal Functional with Many Deviations of Argument. Consider now the problem of extrema of the functional

X!

(14) J(y) = J G(x, y(wo(x), ... , y(wm(x)), y'(wo(x)), ... , y'(wm(x))) dx

xo

with boundary conditions

(15)

(16)

where

y(x) = cp(x) x E [a, xo],

a = min w;(xo) l~i~m

Here y belongs to the space Hp of absolutely continuous functions with derivatives in Lp, 1 ::; p::; oo, with the norm IIYIIHp = max{IIYIIc, IIY'IILp}.

The scalar functions w;(x), i = 1, ... , m, w0(x) = x, w; E C 2[x0 , xi], have their inverses w;1 = /i E C 2 and possess the property

(17) Wj(x) < x, x E [x0,x1], j = l, ... ,m

i.e. they are deviation functions of the retarded type. The boundary function <p E Hp[a, Xo].

The function G is supposed to be twice continuously differentiable with respect to each of its arguments and to be subject to growth constraints similar to (6), which ensures the existence of integrals occurring here.

Define the functions

GJ = G(tj(x), ... , y(wi(tj(x))), ... , y'(w;{tj(x))), .. . )

Put j3j = wj(x1),j = 1, ... ,m; by (17) j3j < x1,j = l. .. ,m. Note that since wj(tj(x)) = x, so y(x) and y'(x) are among the arguments of every GJ. Denote by c;(x) and G~'(x) the partial derivatives with respect to these arguments, correspondingly. Define the function

(19) \It(x, ... , y(w;(lj))), ... , y'(w;(tj(x))), .. . ) := :~::=,;cj, [x]

294 G.A. KAMENSKII

where the summation L.::[x] is meant over those numbers j, for which x < f3i· The following theorem was proved in [11].

THEOREM 3. Suppose that all above mentioned conditions on G and wi(x) are fulfilled. Let an extremum of functional (14) under boundary conditions (15), (16) be attained at y •. Then y. satisfies the equation

X

(20) J 'liy(x) dx- 'liy'(x) = C xo

for almost all x E [xo, x1].

Consider now a moving boundary problem for functional (14), when the point x1 is not fixed, and instead of condition (16) we suppose that y satisfies the condition

(21) (x1, yt) E grafg, Y1 = y(xl),

where g E C1 is a given function. Similarly to Theorem 2, there can be proved the following

, THEOREM 4. Suppose that all assumptions of Theorem 3 are fulfilled, wi(x) < x, j = 1, ... , m andy. is a solution to the moving boundary problem (14), (15), (21). 'Then y. satisfies the transversality condition

(22). [G + (g'- y')Gy']lx~x1 = 0

The simplest moving boundary problem for nonlocal functionals was investigated in [10]. Boundary value problems with fixed boundary values were studied in [11].

REFERENCES

[1] L.E. El'sgol'z, Qualitative Methods of Mathematical Analysis, Trans!. Mathern. Monographs Amer. Mathern. Soc., 12 (1964).

[2] G.A. Kamenskii, Variational and boundary value problems for functionals with deviating argument, Differents. Uravn., 6, 8 (1970), 1359-1358 (In Russian.)

[3] G.A. Kamenskii, On extrema of functionals with deviating argument, Soviet. Math. Dokl., 16 (1975), 1380-1383.

[4] G .A. Kamenskii, On some necessary conditions of extrema of functionals with deviating argument, Nonlinear Anal., TMA, 17, 5 (1991), 457-464.

[5] G.A. Kamenskii, A.D. Myshkis, Variational and boundary value problems for differential equations with deviating arguments, Lecture Notes in Mathematics, 703 (1979), 179-188.


[6] F. Colonius, The maximum principle for relaxed hereditary differential systems with function space and conditions, SIAM J. Contr., Optim., 20 (1982) 695-712.

[7] L.S. Pontryagin, V.G. Boltjanskii, R.V. Gamkrelidze, and E.F. Mischenko, Mathematical Theory of Optimal Processes, Moscow, Fizmatgiz, 1961 (In Russian).

[8] N.N. Krasovskii, Theory of Control of Motion, Moscow, Nauka, 1968 (In Russian). [9] G .A. Kamenskii, Methods of damping of a control system with delay, Theory and

methods of investigation of automatic control systems of nonstationary objects, MAl Press, Moscow, (1985), 46-49 (In Russian).

[10] A.L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhi\user, Basel, 1997. ·

[11] G.A. Kamenskii, Asymmetrical variational problems for a functional with deviating argument, Ukrainian Mathem. J., 41, 5 (1989), 602-609, (In Russian).

[12] G.A. Kamenskii, Nonlinear boundary value problems for functional-differential equations, The Arabian J. for Sc. and Engineering, 17, 4B (1992), 585-589.

[13] G.A. Kamenskii, On extrema of functionals with deviating argument with asymmetric boundary value conditions, Techn. Report, 64, University of Rhode Island, 1991.

[14] G.A. Kamenskii, Boundary value problems for difference-differential equations arising from variational problems, Nonlinear Anal., TMA, 18, 8 (1992), 801-813.


VOLUME 8 2001, NO 3-4 PP. 297-310

INITIAL PROBLEMS FOR HYPERBOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

Z. KAMONT *

Abstract. The Cauchy problem for a quasilinear functional differential equation with unbounded delay is considered on the Haar pyramid. A theorem on the existence and uniqueness of Caratheodory solutions is proved. The method of characteristics and integral inequalities are used.

1. Introduction. Recently numerous papers were published concerning various problems for first order partial functional differential equations considered on the Haar pyramid. The following questions were considered: functional differential inequalities, uniqueness and continuous dependence for initial problems, difference functional inequalities, approximate solutions, existence of classical or generalized solutions. All these problems have the following property: the initial set is bounded. Monograph [3] contains an exposition of recent developments of hyperbolic functional differential equations. Papers [1], [2] initiated the investigations of partial equations with unbounded delay. The theory of ordinary functional differential equations with unbounded delay is presented in monographs [4], [5].

In this paper we start the study of first order partial functional differential equations on the Haar pyramid with unbounded delay. We formulate the problem. For any metric spaces U and W we denote by C(U, W) the class of all continuous functions defined on U and taking values in W. We will use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components. Let B be a Banach space with the norm 11·11· The norm in Rn will also be denoted by 11·11· Let H denote the

• University of Gdansk, Institute of Mathematics, Wit Stwosz 57 222, 80-952 Gdansk, Poland

297

298 Z. KAMONT

Haar pyramid

H = {(t, x) = (t, X1, ... , Xn) E Rl+n : t E [0, a], -b +h(t) :'0 X :'0 b- h(t)}

and

Eo= ( -oo, OJ x [-b, b] c Rl+n

where b = (b1 , . .. ,bn) E R~, R+ = [O,+oo), and h = (hb···,hn) E C([O, a], R~), a > 0. We assume that h is nondecreasing, h(O) = 0 and b- h(a) > 0.

Let X 0 be the space of initial functions w : Eo -+ B. We assume that Xo is a linear space with the norm II · llxa and that (Xo, II · llxa) is a Banach space. For 0 < t :::; a, we put H, = H n ( [0, t] x Rn ). Let II · lit be the supremum norm in the space C(H,, B).

For each t, 0 < t :::; a, we consider the space X, consisting of functions z: Eo U H,-+ B. We assume that H, is a linear space with the norm ll·llx,.

Write X = Xa and II · llx = II · llx. and assume that V :X -+ C(H, B) is a given operator. Let

[!: ([!1, ... , [!n): H x B-+ Rn, f: H x B-+ B, <p: Eo-+ B

be given functions. We consider the quasilinear equation

n

(1) a,z(t,x) + L[!i(t,x,(Vz)(t,x))ax,z(t,x) = f(t,x,(Vz)(t,x)) i=l

with the initial condition

(2) z(t, x) = cp(t, x) on E0 .

We will deal with Caratheodory solutions of problem (1), (2). A function u : Eo U He -+ B, where 0 < c :::; a, is a solution of the above problem provided: (i) ii is continuous on He and the derivatives &,u(t, x), Oxii(t, x) = (Ox, u( t, X), ... , Oxn u( t, X)) exist for almost all ( t, X) E He, (ii) ii satisfies equation (1) almost everywhere on He and condition (2) holds.

We give sufficient conditions for the existence, uniqueness and continuous dependence on initial functions ofCaratheodory solutions of problem (1), (2).

:-·-~·- '.,

HYPERBOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS 299

2. Function Spaces. Denote by CL(Hto B) the class of all w E C(H~o B) such that

Lipw iH, =sup {llw(r, x)- w(r, y)illlx- Yll-1

:

(r,x), (r,y) E Ht,X # y} < +oo.

For a function wE CL(H~o B), we define the norm

llwllt.L = llwllt + Lipw iH,·

The main assumptions on the spaces Xt and the operator V are the following.

AssuMPTION H[X]. Suppose that

1) (Xt, ll·llt)is a Banach space for each t, 0 < t::; a,

2) if z: Eo U Ht --t Band z !Eo E Xo, z iH, E C(Ht, B), then z EXt and

3) (Xo, II · llx,) is a Banach space and the linear subspace Xo.L C Xo is such that Xo.L endowed with the norm II · llxo.L is a Banach space,

4) for each t, 0 < t ::; a, the linear subspace Xt.L c Xt is such that (i) Xt.L endowed with the norm II · iix,.L is a Banach space, (ii) if z : Eo U Ht --t B and z lEo E Xo.L, z IH, E CL{Ht, B), then z E Xt.D

and

iiziix,L :0::: liz IEollxoL +liz iH,iit.L·

Denote by L([t1,t2],R), [t1,t2] C R,theclassofallfunctionsf.l: [t1,t2]--t R, which are integrable on [tl> t2]. Further, we will use the symbol 8 to denote the set offunctions a : [0, a] x R+ --t R+ such that a(·, r) E L([O, a], R+) for all r E R+, a(t, ·) E C(R+, R+), a(t, 0) = 0 and a(t, ·) is nondecreasing for almost all t E [0, a]. Let 8 be the set of functions 1 E C(R.,., R+), which are nondecreasing on R+ and satisfy the condition 1(0) = 0. For (t,x) E [O,a] x [-b,b], we define I[t,x] = {r : (r,x) E Ht} and St = [-b + h(t), b- h(t)]. Further, write B[~<:] = {p E B: IIPII ::; K }, K E R+, and XL = Xa.L·

ASSUMPTION H [VJ. Suppose that the operator V : X --t C ( H, B) satisfies the conditions:

300 Z.KAMONT

1) there isLE R+ such that for z, z EX, (t, x) E H, we have

II(Vz)(t,x)- (Vz)(t,x)ll::; Lllz- zllx,

2) if z E XL, then V z E CL(H, B) and

II(V z)(t, x)- (V z)(t, x)ll ::; l'(llzllt.L) llx- xll, (t, x), (t, x) E H,

where /' E G is independent on z.

REMARK 2.1. It follows from condition 1) of Assumption H[V] that the operator V satisfies the following Volterra condition: if z, z EX, (t, x) E H, and z(r,y) = z(r,y) for (r,y) E H1 then (Vz)(t,x) = (Vz)(t,x).

AssUMPTION H[Q]. Suppose the following:

1) the function g( ·, x,p) : J[a,x] --+ R" is measurable for (x,p) E [-b, b] x B, and 12(t, ·): 81 x B--+ Rn is continuous for almost all t E [0, a],

2) there exists 8 = (81, ... , 8n) E L([O, a], R'j.) such that

le;(t,x,p)l::; 8;(t) for (t,x,p) E H x B, 1::; i::; n,

and t

h(t) ;::: j 8(s) ds, 0

3) there exists fJ E 8 such that

lle(t, x,p)- e(t, x,iJ)II ::0 fJ(t, K) [ llx- xll +liP- Pill

for (t, x), (t, x) E H, p,p E B[K]. Suppose that <p E Xo.L and 0 < c ::; a, d = (do, d1) E R~ and w E

L([O, a], R+)- The symbol Y"'.e[w, d] denotes the class of all functions z: Eo U

He --+ B such that (i) z(t, x) = <p(t, x) on E0 ,

(ii) llz(t,x)ll ::0 do for (t,x) E He, and

r

llz(t,x) -z(r,y)ll ::0 j w(s)ds +d1llx-vll, t

where (t, x), (r, y) E He. For the above <p and z E Y"'.c[w, d], consider the Cauchy problem

(3) r/(r) = g(r,'l)(r), (Vz)(r,'l)(r)) ), 'l)(t) = x,

where (t, x) E He. We consider Caratheodory solutions of (3). Denote by g[z]( ·, t, x) the solution of the above problem. The function g[z] is the characteristic of equation (1) corresponding to z E Y,.e[w, d].

Suppose that Assumptions H[X] and H[VJ are satisfied and <p E X 0 . Let e: Eo u H -t B be the function given by O(t,x) = 0 on Eo u H. Denote by c0 E R+ a constant such that

II(VO)(t,x)ll::; co for (t,x) E H.

It follows from Assumption H[V] that

II(Vz)(t,x)ll::; Lllzllx, +co, (t,x) E H,

where z E X. Write

K = L(II'PIIxo +do)+ co, l£o = II'PIIo.L +do+ d1.

LEMMA 2.2. Suppose that assumptions H[X], H[V], H[o] are satisfied and <p, <jJ E Xo.L, c E (0, a], z E Y,.e[w, d], and z E Yy;.e[w, d].

Then the solutions g[z]( ·, t, x) and g[z]( ·, t, x) are defined on intervals

[0,.\(t,x)] and [0,.'\(t,x)]

such that for T = ,\ ( t, x), 'f = ), ( t, x) we have

(r, g[z](r, t, x)) E fJHc and (T, g[z](f, t, x)) E fJHe,

where fJHe is the boundary of He. Moreover, the solutions are unique and we have the estimates

(4) llg[z](r, t, x)- g[z](r,t, x)ll ::;

where T E [0, min{,\(t, x), .\(t, x) }] and

T

(5) llg[z](r,t,x) -g[z](r,t,x)ll::; L j fJ(s,R) [II'P-<PIIx,+ t

302 Z. KAMONT

+ll(z-z)[H,[[s] ds[ exp{(l+/(1\;o))! f3(s,Ft)ds }'

where T E [0, min{.\(t, x))(t, x) }.

Proof. For z E Y<p.e[w, d], we have

(6) [[(Vz)(t,x)[[::;L([[rp[[x0 +do)+eo, (t,x)EH

and

(7) [[z[[t::; [[rp[[o.L +do+ d1, t E [0, c].

The existence and uniqueness of the solutions of problem (3) follows from classical theorems. Note that the right-hand side of the differential system satisfies the Caratheodory assumptions, and the Lipschitz condition

lle(t, y, (V z)(t, y )) - e(t, y, (V z)(t, y)) II ::; (3(t, Ft) (1 + 1(1\;o)) [[y- iii I

holds on He. The function g[z]( ·, t, x) satisfies the integral equation

T

g[z]( T, t, x) = x +I e(s, g[z](s, t, x), e(s, t, (V z)(s, g[z](s, t, x))) ds, t

where (t, x) E He, T E [0, ,\(t, x)]. It follows from assumptions H[V] and H[e] that the integral inequality

t

[[g[z](T, t,x)- g[z](T, f,x)[[::; [[x- x[[ + Itt o(s) II ds + t

r

. +I (3(s,Ft)(l+!(ii;o)) [[g[z](s,t,x)-g[z](s,f,x)[[ds t

is satisfied for (t, x), (f, x) E He, t E [0, min{.\(t, x), ,\(t, x) }]. Now we obtain ( 4) by the Gronwall inequality.

For z E Y<p.e[w, d] and z E Y.;;.c[w, d], we have the estimate

(8) II (V z)(s, g[z](s, t, x))- (Vz)(s, g[z](s, t, x)) II ::;

::; !(Ko) [[g[z](s, t, x) - g[z](s, t, x) II + L ([['P- <Pllxo + II (z- z)[H, lis)·

It follows from assumptions H[12], H[X] and H[V] that the integral inequality

llg[z](r, t,x)- g[z](r, t,x)ll :S

T

:S L I (3(s, R) [ II'P- <PIIxo + ll(z- z)IH,IIs l ds + t

T

+(1 + 'Y( "o)) I (3( s, R) II g[z]( s, t, x) g[z]( s, t, x) II ds t

is satisfied, where (t, x) E He, t E [0, min {>.(t, x))(t, x) }]. Now we obtain (5) by the Gronwall inequality. This completes the proof of the lemma.

3. Existence and Uniqueness of Solutions. We construct an integral operator corresponding to problem (1), (2). Suppose that assumptions H[X], H[12], H[V] are satisfied and c E (0, a], z E Y"'.c[w, d]. Define the operator U"' for all z E Y"'.c[w, d] by the formulas (9)

t

U"'z(t, x) = <p(O, g[z](O, t, x)) + If( s, g[z](s, t, x ), (V z) (s, g[z](s, t, x))) ds 0

where (t,x) E He and

(10) U"'z(t, x) = <p(t, x) on E0 .

The operator U"' is obtained by integration of equation (1) along characteristics. We give sufficient conditions for the existence and uniqueness of a solution of the equation z = U"'z on the space Y"'.c[w, d].

AssUMPTION H[F,<p]. Suppose that

1) the function f( · ,x,p): J[a,x]--+ B is measurable for (x,p) E [-b,b] x B and f(t, ·) : S1 xB --7 B is continuous for almost all t E [0, a],

2) for (t,x,p) E H x B[,;], we have llf(t,x,p)ll :S a(t,K) and

llf(t, x,p)- f(t, x,fi)ll :S f3(t, ,;) [ llx- xll +liP- vii],

where (t,x,p), (t,x,p) E H x B[,;],

3) there are L, L 0 E R+ such that

II'P(O, x)ll :S L and li'P(O, x)- <p(O, y)ll :S Lo llx- vii,

304 Z. KAMONT

where x, y E [-b, b].

REMARK 3.1. For simplicity of notations, we have assumed that (!

and f satisfy the Lipschitz condition on the space H x B[~~:] with the same coefficients.

LEMMA 3.2. Suppose that assumptions H[X], H[g] and H[f, <p] are satisfied and <p E Xo.L· Then there are d = (do, d1) E R~, c E (0, a] and wE L([O, c], R+) such that U"' : Y<p.e[w, d]-+ Y<p.e[w, d].

Proof. Suppose that d = (do, dJ), c E (0, a] and w E L([O, c], R+) satisfy the conditions:

(11)

(12)

where (13)

e

do ~ L + j a(s, f£) ds, d1 ~De, 0

w(t) ~De llo(t)ll + a(t, f£),

De= [ Lo + (1 + 'Y(~~:o)) l f3(s, f£) ds] exp [ (1 + 'Y(Ko)) l f3(s, f£) ds] .

Suppose that z E Y<p.e[w, d]. From assumptions H[f,<p] and from (6) it follows that

e

(14) IIU'Pz(t, x)ll ~ L + j a(s, f£) ds ~ do, (t, x) E He. 0

Assumption H[f,<p] and (7) imply that for the above z and for (t, x), (l, x) E He we have

II U"'z(t, x) - U"'z(l, x) II ~ II'P(O, g[z](O, t, x))- <p(O, g[z](O, l, x))ll+

t

+ j II f(s, g[z](s, t, x), (V z)(s, g[z](s, t, x)) )-o

t

-f(s,g[z](s,l,x), (Vz)(s,g[z](s,l,x))) lids+ j a(s,f£)ds t

Then, using Lemma 2.2 and Assumption H[V] we obtain

IIU'Pz(t, x)- U'Pz(t, x)ll :::; De [ llx- xll + I ll8(s) II ds ] + f

j a(s,R:)ds t

and consequently

t

(15) IIU'Pz(t,x) U'Pz(t,x)ll:Sdlllx-xll+ jw(s)ds t

It follows from (14), (15) that U'Pz E Y'P_c[w, d], which completes the proof of the lemma. D

THEOREM 3.3. Suppose that assumptions H[X], H[V], H[a] and H[j, 'P] are satisfied and <p E Xo.L· Then there exist d = (do, d1) E R~, c E (0, a] and w E L([O, c], R+) such that problem (1), (2) has exactly one solution U E Y<p.c[w, d].

If <P E Xo.L satisfies condition 3) of assumption H(j,<p) and u E Y.;;.c[w, d] is a solution of equation (1) with the initial condition z(t, x) = <P(t, x) on E0 ,

then there is Ac E R+ such that (16) II (u- u)IH, lit::; At [ II'P- <PIIxo +max{ jj<p(O, y)- <P(O, y)JI: Y E [-b, b]} ].

where t E [0, c].

Proof. Let d = (d0 , d1) E R~, c E (0, a], and let w E L([O, c], R+) be such that conditions (11), (12) are satisfied. Lemma 3.2 shows that U'P : Y<p.c[w, d]-t Y<p.c[w, d]. Put

c

qc + (1 + Dc)L j (3(s, R:) ds, 0

where De is given by (13) and assume that c is such a constant that qc < 1. Now we prove that the operator U'P is a contraction on Y'P.c[w, d].

It follows from assumptions H[X] and H[V] that for z, z E Y<p.c[w, d] we have

II (Vz)(s,g[z](s,t,x))- (Vz)(s,g[z](s,t,x)) II::;

::; Lll (z - z) In,! I, + f'(Ko) II g[z] (s, t, x) - g[z](s, t, x) II·

306 Z. KAMONT

From Lemma 2.2 and from (13) it follows that

II (U<Pz)(t,x)- (U<Pz)(t,x) II::; Lollg[z](O,t,x) g[z](O,t,x)ll+

t

+ j f3(s, K) [ llg[z](s, t, x)- g[z](s, t, x)ll +II (V z)(s, g[z](s, t, x))-o

e

- (Vz)(s,g[z](s,t,x))IIJ ds :S L(l+De) j f3(s,K)dsll(z-z)IH,IIt, (t,x) E He 0

and consequently

By the Banach fixed point theorem, there exists a unique solution u E

Y<P.e[w, d] of the equation z = U<Pz. The function u is a solution of problem (1), (2). This property of u can be proved using methods from [3], Chapter 2.

Now we prove relation (16). If u = U<Pu and u = Uq;u, then we have for (t,x) E He

llu(t, X) - u( t, X) II ::; llrp(O, g[u](O, t, X)) - r,O(O, g[ii] (0, t, X)) II+

t

+ f II f(s, g[u](s, t, x), (Vu)(s, g[u](s, t, x)) )-o

- f(s, g[ii](s, t, x), (Vu)(s, g[u](s, t, x))) II ds::;

::; Lollg[u](O,t,x) g[u](O,t,x)ll +

+max{ llrp(O, y)- rp(O, y)il : y E [-b, b] }+

t + f f3(s, K) [ llg[u](s, t, x) - g[u](s, t, x) II + II (Vu) (s, g[u](s, t, x))-

0

-(Vu)(s,g[u](s, t,x))IIJ ds::;

:S max{ llrp(O, y) r,O(O, Y)ll: Y E [-b, b]} + qeii'P- <PIIxo+

t +L(1 + Delf f3(s, K) ll(u- ii)IHJs ds.

0

Put '1/J(t) = ll(u- u) IH,IIt. t E [0, c]. It follows from the above estimates that '1/J satisfies the integral inequality

'1/J(t) :S max{ llrp(O, y) - r,O(O, y) II : Y E [ -b, b]} + qeii'P- <PIIxo+

:-·-~·· - .,

t

+L(l +De) j f3(s, k) '1/J(s) ds, t E [0, c]. 0

It follows from the Gronwall inequality that we have estimate (16) with

This completes the proof of the theorem. D

4. Phase Spaces. We give examples of spaces X, satisfying assumption H[X].

EXAMPLE 4.1. Let X 0 be the class of all functions w: Eo -+ B, which are uniformly continuous and bounded on E0 . For w E Xo, we write

[[wllxo =sup{ [[w(t, x)[[ : (t, x) E Eo}.

Let Xo.L C X 0 denote the set of all wE X 0 such that

Lipw [Eo =sup {llw(t, x)- w(t, x)[[· [[x- x[[-1 :

(t, x), (t, x) E Eo, xI x} < +oo.

Write [[wllxoL = [[w[[x0 + Lipw [Eo· Let X 1, 0 < t ::; a, be the set of all functions z : Eo U H1 -+ B such that

z [Eo E Xo and z [H, E C(H1, B).

For z E X,, we put

[[z[[t =sup{ [[z(t,x)[[: (t,x) E Eo U H, }.

Let Xt.L c X, denote the set of all z E X, such that

z lEo E Xo.L and z [H, C CL(H, B).

For z E Xt.L, we put

[[zllx,L =sup{ [[z(s, y)[[ : (s, y) E Eo U H, }+

+sup { [[z(s,x)- z(s,x)[[[[x- .r[[- 1: (s,x), (s,x) E E0 U H,, xI x}.

308 Z.KAMONT

Then assumption H[X] is satisfied.

EXAMPLE 4.2. Let 1j;: (-oo,OJ-+ (O,oo) be a continuous and nonincreasing function. Let X 0 be the space of continuous functions w : Eo -+ B such that

lim llwJ~~)II = 0, x E [-b,b]. t-t-00 t

Put

{ llw(t,x)ll }

llwllxo =sup 1/;(t) : (t, x) E Eo .

Denote by Xo.L c Xo the set of all w E X 0 such that

Lipwi.P.Eo =sup { llw(t,x)- w(t,x)ll ( 1/;(t) llx- xll t 1:

(t,x) (t,x) E Eo, X 1 x} < +oo.

For wE Xo.L, we put llwllo.L = llwllxo + Lipwi,;;.Eo· Let Xt, 0 < t ::; a, be the set of all z : Eo U Ht -+ B such that

We define the norm in the space Xt by

llzllx, = llwllxo + llvllt. where w = z lEo, v = z Is,.

Let Xt.L c Xt denote the set of all z E Xt such that

For z E Xt.L, we put

llzllx,.L = llzllx, + Lipw I.P.Eo+

+sup { llz(r,x)- z(r,x)llllx- xll-1: (r,x), (r,x) E Ht, X 1 x},

where w = z lEo· Then assumption H[X] is satisfied.

EXAMPLE 4.3. Let r E R+ and p 2 1 be fixed. Denote by X 0 the class of all functions w : Eo -+ B such that

(i) w is continuous on [-r, OJ x [-b, b], (ii) for x E [ -b, b], we have

-r

j llw(s,x)IIPds < +oo, -oo

(iii) w(t, ·): [-b, bJ-+ B is continuous for almost all t E ( -oo, -rJ. Write

llwllxo =sup { llz(t, x)ll : (t, x) E [-r, OJ X [-b, b]} +

Let Xo.L C X 0 be the set of functions w E X 0 such that

LiP[r,pJ w =sup { llw(t, x)- w(t, x)llllx- xll-1 :

(t, x), (t, x) E [-r, OJ X [-b, bj, x # x} +

+sup { llx ~ xll (_[ llw(s, x) - w(s, x)IIP ds) ~ x, x E [ -b, bj, x # x} < +oo.

For wE Xo.L, we put llwllo.L = llwllxo + LiP[r,p] w. Let Xt, 0 < t ::; a, be the set of all z : E0 U H, -+ B such that z lEo E X 0

and z IH, E C(H,, B). The norm in the space X, is defined by llzllx, = llwllxo + llvllt, where w = z IE" v = z IH,·

Let Xt.L c X, denote the set of all z E X, such that z lEo E Xo.L and z IH, E CL(Ht, B). For z E xt.L, we put

llzllx, L = llzllx, + LiP[r,p] W

+sup { llz(r, x)- z(r, x)llllx- xll- 1: (r, x), (r, x) E H,, x # x},

where w = z lEo· Then assumption H[XJ is satisfied.

REMARK 4. 4. Differential equations with a deviated argument and differential integral equations can be derived from (1) by specializing the operator V.

310 Z. KAMONT

REFERENCES

[1] T. Czlapinski, On the mixed problem for quasilinear partial functional differential equations with unbounded delay, Ann. Polan. Math., to appear.

[2] Z. Kamont, Hyperbolic functional-differential equations with unbounded delay, Zeit. Anal. Anwend., 18 (1999), 97-109.

[3] z. Kamont, Hyperbolic Functional Differential Inequalities and Applications, Kluwer Acad. Pub!., Dordrecht, Boston, London, 1999.

[4] Y. Hino, S. Murakami, and T. Naito, Functional differential equations with infinite delay, Lect. Notes Math., 1473, 1991.

[5] V. Lakshmikantham, Wen Li Zhi, and Zhang Bing Gen, Theory of Differential Equations with Unbounded Delay, Kluwer Acad. Pub!., Dordrecht, Boston, London, 1994.


VOLUME 8 2001, NO 3-4 PP. 311-322

PASSAGE TO THE LIMIT IN QUASILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS ON A PLANE

V.L. KAMYNIN *

1. Introduction. In this paper we investigate the question of convergence of the solutions um(t, x) of the problem

(1.2)

(1.3)

t

Ut = am(t,x,u,ux)Uxx + j bm(t,r,x,u(r,x),ux(r,x))uxx(r,x)dr+ 0

(t,x) E Q =: [O,TJ x [0,1], m= 1,2, ...

u(O,x)=O, xE[O,I],

u(t, 0) = u(t, l) = 0, t E [0, T],

to the solution u00 (t,x) of the "limit" problem (1.100), (1.2), (1.3), which is

characterized by the symbol "oo" in the coefficients of the equation. We suppose that the functions am(t, x, u, p) and dm(t, x, u, p) converge weakly in L2 (Q) to the functions a00 (t,x,u,p) and d00 (t,x,u,p) respectively, and that the functions bm(t, T, x, u,p) converge weakly in L2 ([0, T] x Q) to the function b00 (t, T, X, u,p).

This setting is closely connected to the mathematical simulation of various physical processes in strongly inhomogeneous materials with memory.

The questions of global solvability of the problems of type (l.lm), (1.2), (1.3) were investigated by different methods in the papers of several authors.

' Moscow State Engineering Physics Institute, Kashirskoe shosse, 31, 115409 Moscow

311

312 V.L. KAMYNIN

We use papers [1,2] for the existence results and some a priori estimates of the solutions, which do not depend on the smoothness properties of the coefficients of equations ( 1.1"').

Concerning the investigation of the problems of passage to the limit in differential equations with weakly converging coefficients, it should be mentioned that such investigation was started in papers [3-5] in connection with the development of G-convergence theory and theory of homogenization. A sufficiently complete theory for linear and quasilinear differential equations of parabolic type was constructed in [6-9, etc.]. Note that the consideration of integro-differential parabolic equations is closely connected with some settings of inverse problems and the questions of passage to the limit in such problems that were studied in [10,11] (see also [12]).

In papers [6-12] one can find minimal additional requirements on weakly converging coefficients of the equations considered, which provide the possibility of passage to the limit in corresponding equations. The form of these conditions turns out to be sufficiently general. Moreover, for some equations such conditions appear to be not only sufficient but also necessary.

In section 2 of the present paper we establish theorems on passage to the limit in the sequence of problems (1.1"'), (1.2), (1.3), m = 1, 2, ... , oo, under additional assumptions on the coefficients a"'(t,x,u,p) and bm(t,r,x,u,p) similar to the ones from the papers [6-12]. In section 3 we prove that these conditions appear to be also necessary for the possibility of passage to the limit in problems (l.lm), (1.2), (1.3) (theorems 3.1 and 3.2). Moreover, in theorem 3.1 one proves simultaneously that the additional conditions on the coefficients am(t,x,u,p) in the sequence of parabolic equations

Ut = am(t, X, U, Ux)Uxx + b"'(t, X, U, Ux), (t, x) E Q,

obtained in the papers [6,7] are also not only sufficient but necessary for the possibility of passage to the limit in the boundary value problems for equations (1.4m).

We introduce following notations. We put

Q = [0, Tj X [0, T] x [0, l].

The Lebesgue spaces L2 ([0, l]), L 2 (Q), L 2 (Q) and Holder spaces C0•"'( Q),

C1•"'( Q), C2•"'( Q) with corresponding norms are understood in usual sense (see [13]).

QUASILINEAR PARABOLIC INTEGRO--DIFFERENTIAL EQUATIONS 313

In the sequel we assume that the following conditions are satisfied.

A. The functions am(t,x,u,p),bm(t,T,X,u,p),dm(t,x,u,p) are measurable and uniformly bounded with respect to m for the bounded variables u,p.

B. There exist positive constants a0 , db b1 such that for ( t, x) E Q, T E [0, T], u E R\ p E R 1 uniformly with respect to m = 1, 2, ... , oo we have

(1.5)

(1.6)

(1. 7)

C. For every fixed m =I oo, the coefficients of equations (1.1m) have continuous partial derivatives up to the second order and satisfy compatibility conditions for x = 0 and x = l up to the second order (see [14]).

D. There exist modulus of continuity w(p) and constants L > 0, ( E (0, 1) such that for any h1 , h2 E R 1 , uniformly with respect to m = 1, 2, ... , oo, (t, x) E Q, T E [0, T], u E [-K1 , K1],p E [-K2, K2] (where Kb K2 are defined below in (1.8),(1.9)) one has

lam(t, x, u + h1,P + h2)- am(t, x, u,p)l + lbm(t, T, x, U + h1,p + h2)-

DEFINITION. By generalized solution of problem (1.1m), (1.2), (1.3), m = 1, 2, ... , oo, we mean a function um(t, x) E C 0•"( Q) (for some a E (0, 1) ), u:;'(t,x) E L00 (Q), u;"(t,x), n::'x(t,x) E L2 (Q), which satisfies equation (l.lm) almost everywhere in Q and satisfies boundary conditions (1.2),(1.3).

One has

THEOREM 1.1 [2]. Let conditions A - C be satisfied. Then, for any fixed m = 1,2, ... , there exists a unique solution um(t,x) of problem (l.lm), (1.2), (1.3), and the uniform estimates with respect to m

(1.8)

314 V.L. KAMYNIN

(1.9)

(1.10)

hold with positive constants Kl> K2, K3, depending only on l, T, a0 , d1 , b1 .

By virtue of estimates (1.8)-(1.10) and embedding theorem from [15, p. 430], there exists a E (0, 1) such that um(t, x) E C 0·"(Q), and a uniform estimate with respect to m

(1.11)

is fulfilled, where K 4 depends on the same quantities as Kl> K2, K3. Because of estimates (1.8)-(1.11) one can assert that there exist a func

tion v(t,x) E C0·"(Q), vx(t,x) E L00 (Q), Vxx(t,x),vt(t,x) E L2(Q) and a subsequence mv -+ oo such that uniformly on Q

(1.12) um"(t,x) ==} v(t,x),

in the norm of L2(Q)

(1.13) u;;'"(t,x)-+ vx(t,x),

weakly in L2 ( Q)

(1.14) u:;:;(t,x)--'- Vxx(t,x), u7'"(t,x)--'- Vt(t,x),

and the function v(t, x) satisfies estimates (1.8)-(1.11) with the same constants.

2. Passage to the Limit in Problem (1.1 m), (1.2), (1.3). In theorem 2.1 proved below, we give the condition under which the function v(t, x) defined by relations (1.12)-(1.14) is a generalized solution of problem (1.100

), (1.2), (1.3), and it is hence possible to pass to the limit (generally speaking, up to a subsequence) in problem (1.1m), (1.2), (1.3).

THEOREM 2.1. Let conditions A-D be satisfied. Suppose that as m-+ oo for any fixed u E [-K1,KI],p E [-K2 , K 2] (where K1 is from (1.8), and K2 is from (1.9)) (2.1) am(t,x,u,p)--'- a00 (t,x,u,p), dm(t,x,u,p)--'- d00 (t,x,u,p) weakly in L2(Q)

and

(2.2) bm(t,r,x,u,p)--'- b00 (t,r,x,u,p) weakly in L2 ([0,T] x Q).

~ ·-·-~·- ' .,

QUASILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS 315

Suppose that for any u(t,x) E C0·"'(Q),ux(t,x) E L 00 (Q) there exists a sequence ~t(m) ---+ 0 as m---+ oo (generally speaking, depending on u(t, x)) such that for all g(t,x) E C00 (Q) compactly supported with respect to x, we have

(2.3)

T I I l[am(t, x, u, Ux)- a00 (t, x, u, Ux)]gxx(t, x) dxdt ~ 0 0

~ ft(m) {lglco,"(Q) + IIYxiiL=(Q) + IIYxxiiL,(Q) + llgtiiL,(Q)} and for all 'lj;(t, T, x) E C00 (rl) compactly supported with respect to x, we have

(2.4)

T t I I I l[bm(t,r,x,u(r,x),ux(T,X))o 0 0

-b00 (t, T, x, u( T, x), Ux( T, X) )]'¢xx(t, T, x) dxdrdtl ~

~ ~t(m){l'¢lco,o(f!) + ll'¢xiiL=(f!) + ll'¢xxli£,(n)+

+111/JtxxiiL,(f!) + ll'¢tli£,(n) + ll'¢riiL,(f!)}· Then the function v(t, x) defined above by relations (1.12)- (1.14) is a generalized solution of problem (1.1 00

), (1.2), (1.3).

Proof. Let us show that the function v(t, x) defined by relations (1.12)(1.14) satisfies equation (1.100

) almost everywhere in Q. Let <p(t, x) be an arbitrary but fixed function from Ci)"(Q). We consider

the quantity

T I t

J = I 1[-Vt + a00 (t, x, v, Vx)Vxx +I b00 (t, T, x, v(r, x), Vx(T, x))vxx(r, x) dr+ 0 0 0

+d00 (t, x, v, Vx)]<p(t, x) dxdtl.

As the function um"(t, x) satisfies equation (l.lm"), then for any v = 1,2, ...

T I

Tl

J ~ I I ( -v1 + n';'" )'P dxdt + 0 0

+I I [a00 (t,x,v,vx)Vxx- am"(t,x,nm",n';'")n';:;]'Pdxdt + 0 0

316 V.L. KAMYNIN

T t l

+I I I IW"(t, T, x, v(T, x), Vx(T, x))vxx(T, x)o 0 0

-bm" (t, T, x, um" ( T, x), u;;'" ( T, x))u;;'; ( T, x)]<p(t, x) dTdtdxl+

Tl

+ ll[d00(t,x,v,vx) -dm"(t,x,um",u;;'")]<pdxdt-0 0

(2.5)

Now we show that for any c: > 0 and for sufficiently large v,

(2.6) 11 < c:, J~ < c:, Jf < c:, J~ < c:.

For Jj, estimate (2.6) follows directly from (1.14). In order to estimate J~, we write the inequality

By virtue of condition D,

and so, taking into account the limit relation (1.13), we find that, for sufficiently large v, the quantity r1 < c:j3. In the same way we obtain from conditions D and (1.12) the estimate r2 < c:/3 for sufficiently large v. Finally, the term r3 tends to zero as v -+ oo by virtue of relation (2.1) and lemma from [7].

Thus, for sufficiently large v, one obtains (2.6) for the integral J:{. Now we obtain the same estimate for Jfl. We have

T l t

Jf ::0 Ill [bm"(t,T,X,Um",u::")- bm"(t,T,x,v,vx)]u::; dT · <pdxdt + 0 0 0

(2.8)

T l t +Ill [bm"(t,T,x,v,vx)- b00 (t,T,x,v,vx)]u~; dT · <pdxdt + 0 0 0

T l t +Ill (u~;(T,x)- Vxx(T,x))b00 (t,T,x,v,vx)'P(t,x)dTdxdt -0 0 0

=I[+ I~+ If.

The term I'{ is small for sufficiently large v due to condition D and relations (1.12),(1.13). This fact can be proved in a similar way as for the integrals r 1

and r2 in inequality (2.7). The term I!{--+ 0 as v--+ oo due to limit relation (1.14).

Thus, for sufficiently large v,

(2.9) I'{+ If < 2c/3.

Finally, we transform the term I~ in the following way:

T l t

I~ ::0 Ill [bmv (t, T, X, V, Vx)- b00 (t, T, X, V, Vx)] X

0 0 0

T l t +2111 [bm"(t,r,x,v,vx)- b00 (t,r,x,v,vx)] X

0 0 0

318 V.L. KAMYNIN

T l t

+2 I I I [bm"(t,r,x,v,vx)- b00 (t,r,x,v,vx)]vx(r,x)cpx(t,x)drdxdt + 0 0 0

T l t +I I I [bm"(t,r,x,v,vx) 0 0 0

-b00 (t, T, X, V, Vx)] ( Um" (r, x) - v( T, X) )'Pxx(t, x) drdxdti +

T l t +I I I [bm"(t,r,x,v,vx)- b00 (t,r,x,v,vx)]v(r,x)tpxx(t,x)drdxdt 0 0 0

(2.10)

The termS) in the right-hand side of inequality (2.10) can be estimated for sufficiently large v in the following way:

(2.11) + llum"(r,x)cpt(t,x)IIL,(li) + llu:""(r,x)cp(t,x)IIL,(n)} < c/15.

by virtue of (2.4) via estimates (1.8)-(1.11).

On the other hand, for sufficiently large v

(2.12) s~ + sr < 2c/15

due to relations (1.12), (1.13), and

(2.13) S!) + S~ < 2c/15

by virtue of condition (2.2) of the theorem. Substituting (2.11)-(2.13) into (2.10) and taking into account (2.9), one

gets that the integral J'3 satisfies estimate (2.6) for sufficiently large v. The same estimate for the integral J!f. is proved in a similar way because

of conditions (2.1) and (2.3).


Since c > 0 is arbitrary and the space C8"(Q) is dense in L2 (Q), it follows from estimates (2.6) and inequality (2.5) that the function v(t, x) satisfies equation (1.1 "") almost everywhere in Q.

It is obvious from (1.12) that v(t, x) satisfies boundary conditions (1.2), (1.3). Thus, the function v(t, x) is a generalized solution of problem (1.1""), (1.2), (1.3).

Theorem 2.1 is proved. 0

COROLLARY 2.1. Under the conditions of theorem 2.1, one obtains the following limit relations: (2.14)

am" (t, x, um" (t, x), u;;'" (t, x) )u;;'; (t, x) -' a""(t, x, v(t, x), Vx(t, x))vxx(t, x),

(2.15) t

t J bm" (t, r, x, um"(r, x), u;;'" (r, x))u;;'; (r, x) dr-' 0

-' J b""(t,r,x,v(r,x),vx(r,x))vxx(r,x)dr weakly in L2 (Q) as v-+ oo. 0

Proof. Relations (2.14),(2.15) follow directly from estimates (2.6) for J!f. and Jf. o

3. On the Necessity of the Additional Conditions. In this section we show that condition (2.3) is necessary in a certain sense for the possibility of passage to the limit in problem (l.lm), (1.2), (1.3).

Consider in Q the first boundary value problem for the parabolic equation

Ut = am(t, X)Uxx + dm(t, x), m = 1, 2, ... , 00,

with initial and boundary conditions (1.2), (1.3). Assume that the coefficients of equations ( 3.1 m) are measurable, bounded

in Q uniformly with respect tom, am(t, x) 2: a0 > 0, (t, x) E Q, and

(3.2)

weakly in L2 (Q) as m-+ oo. One has

THEOREM 3.1. Assume that for all sequences of functions dm(t, x), possessing the properties listed above, for the sequences of solutions um(f;, x) E

320 V.L. KAMYNIN

Wi'2(Q) of corresponding problems (3.1m), (1.2), (1.3) (existense and uniqueness of such solutions are guaranteed, for example, by the results of [13]) as m -+ 00 the following limit relations hold:

(3.3) um(t, x) ==? U00 uniformly on Q,

(3.4) u;:(t,x)-+ u';'(t,x) in the norm of L2 (Q),

(3.5) u;:x(t,x)---' u:(t,x), u;"(t,x)---' ur:'(t,x) weakly in L 2(Q),

(see (2.14)}. Then there exists a sequence J.L(m) -+ 0, m-+ oo, such that for any g(t,x) E C00 (Q) compactly supported with respect to x,

(3.7) I/ £[am(t, x) - a00 (t, x)]gxx dxdtl :'::

:':: f.L(m) {lglco.o(Q) + llgxiiL=(Q) + llgxxiiL,(Q) + llgtiiL,(QJ} ·

Proof. We consider the sequence of equations (3.1m) with dm(t,x) -am(t, x)- a00 (t, x). Then d00 (t, x) _ 0, and hence u00 (t, x) - 0 in Q.

For solutions um(t,x) of problems (3.1m), (3.2), (3.3) with the indicated dm(t,x), we obviously have

l

~ j (u;:(T, x)) 2 dx + j J am(t, x) (u:,(t, x))

2 dxdt =

0 Q

= j £[a00 (t, x)- am(t, x)]u;:x(t, x) dxdt

This relation implies inequality

l

~ j (u;:(T, x))2

dx + aollu:,IILcQJ :':: 0

Since u00 (t, x) = 0, then the first term on the right-hand side of (3.8) tends to zero as m --t oo via (3.5), while the second term tends to zero as m --t oo due to (3.6).

Thus, (3.8) implies

(3.9) llu:;'(T, ·)IIL,([o,t]), llu;;:,IIL,(Q) --> 0, m--> oo.

Let g(t,x) E C00 (Q) be an arbitrary function compactly supported with respect to x. We multiply equation (3.1m) by 9xx(t,x) and integrate over Q. As a result, we obtain the equality

J £[am(t,x)- a00 (t,x)]9xx(t,x) dxdt =

= J k ur;'(t, x)gxx(t, x) dxdt- J k am(t, x)u;;:,(t, x)gxx(t, x) dxdt =

l

= - J u;;'(T, x)gx(T, x) dx- J k u;;'x(t, x)g1(t, x) dxdto

-J k am(t, x)u;;:,(t, x)gxx(t, x) dxdt,

which implies the inequality

(3.10)

I/ £[am(t, x)- a00 (t, x)]9xx dxdtl :<;

:'0: (11u:;'(T, x) li£2 ([0,t]) + llu;;:, li£2 (Q) + s~p lam ( t, x) I · llu;;:, li£2(Q)) X

X (ll9x(T,x)ll£,([o,t]) + ll9tiiL,(Q) + ll9xxiiL,(Q)) =

= Jl.(m) (ll9x(T, x)li£,([0,1]) + ll9ti1Lz(Q) + ll9xxli£,(Q)) · By virtue of limit relations (3.9), one gets Jl.(m) --t 0 as m --t oo, and

hence (3.10) implies (3.7). The theorem is proved. D

REMARK 3.1. Since equation (3.1m) is a usual differential parabolic equation, theorem 3.1 implies the necessity of additional conditions in theorems on passage to the limit in nondivergent parabolic equations considered in [6,7]. Note that the necessity of similar additional conditions for parabolic differential equations with divergent main part was proved previously in [9].

By the same arguments, one proves that condition (2.4) is also necessary for passage to the limit in problem (l.lm), (1.2), (1.3).

322 V.L. KAMYNIN

REFERENCES

[1] H.M. Yin, The classical solutions for nonlinear parabolic integrodifferential equations, J. Integr. Equat. Appl., 1, 2 (1988), 249-263.

[2] J.R. Cannon andY. Lin, A remark on the solvability for nonlinear parabolic integradifferential equations in one-dimensional spaces, Boll. Unione Matern. !tal., Ser 7, 5-B (1991), 853-874.

[3] P.K. Senatorov, Coefficient stability of the solutions of second order ordinary differential equations and of parabolic equations on the plane, Differentsial 'nye Uravneniya, 7, 4 (1971), 754-758 (in Russian).

[4] V.G. Markov and O.A. Oleinik, On propagation of heat in one-dimensional disperse media, Prikl. Mat. Mekh., 39, 6 (1975), 1073-1081 (in Russian).

[5] V.V. Zhikov, S.M. Kozlov, and O.A. Oleinik, Hornogenizaton of differential operators, Nauka, Moscow, 1993.

[6] S.N. Kruzhkov and V.L. Kamynin, Convergence of the solutions of parabolic equations with weakly converging coefficients, Dokl. Akad. Nauk SSSR., 270, 3 ( 1983)' 533-536.

[7] S.N. Kruzhkov and V.L. Kamynin, On passage to the limit in quasilinear parabolic equations, Trudy Matern. Inst. irn. V.A.Steklova, 167 (1985), 183-206 (in Russian).

[8] V.L. Kamynin, On passage to the limit in quasilinear elliptic equations with several independent variables, Matern. Sb., 174, 1 (1987), 45-63 (in Russian).

[9] V.L. Kamynin, Limit passage in quasilinear parabolic equations with weakly converging coefficients, and the asymptotic behavior of solutions of the Cauchy problem, Matern. Sb., 181, 8 (1990), 1031-1047 (in Russian).

[10] V.L. Kamynin, Passage to the limit in the inverse problem for nondivergent parabolic equations with a final overdetermination, Differentsial 'nye Uravneniya, 28, 2 (1992), 247-253 (in Russian).

[11] V.L. Kamynin, On convergence of the solutions of inverse problems for parabolic equations with weakly converging coefficients, Elliptic and Parabolic Problems, Pitman Research Notes in Mathematics, 325 (1995), 130-151.

[12] V.L. Kamynin and LA. Vasin, Inverse problems for linearized Navier-Stokes equations with integral overdetermination. Unique solvability and passage to the limit, Ann. Univ. Ferrara, Ser. VII- Sc. Mat., 38 (1992), 229-247.

[13] S.N. Kruzhkov, Quasilinear parabolic equations and systems with two independent variables, Trudy sern. im. I.G.Petrovskogo, 5 (1979), 217-272 (in Russian).

[14] A. Fridman, Partial differential equations of parabolic type, Prentice-Hall, 1964. [15] O.V. Besov, V. Il'in, and S.M. Nikolskii, Integral representations and embedding

theorems, Nauka, Moscow, 1975 (in Russian).

VOLUME 8 2001, NO 3-4 PP. 323-333

SOME ALGEBRAS OF OPERATORS MAJORIZED BY A CONVOLUTION *

V.G. KURBATOV t

Abstract. Let IE be a Banach space with the norm I · I and N1 be the algebra of all linear bounded operators N : Lp(JR:', IE) __, Lp(IR:', IE) satisfying the estimate I (N x) (t) I ::; J (3(t- s)lx(s)l ds for some non-negative function (3 E L1 • If IE is finite-dimensional, it is

R< shown that the operator N is integral. The main result: if an operator 1 + N, N E N1 , is invertible, then the inverse operator (1 + N)-1 has the form 1 + M, where ME Nr.

1. The Formulation of the Main Result. We consider the group lR', c EN, and its subsets IQ!k = k + [0, 1]< and 1[1 = k + (0, 1]', k E Z'.

Let E be a complex Banach space with the norm 1·1· For any function x : JR' -t E, we consider the family { Xk : k E zc}, where Xk is the restriction of x to IQ!k. Let 1 ::; p, q ::; oo. We define Lpq = Lpq(lR', E) as the space of all classes (with the identification a.e.) of measurable functions x : JR' -t E, for which each function .Tk : 1\l!Tc -t E, k E Z', belongs to Lp(l\l!k, E) and the family { xk} belongs to lq(Z', Lp(IQ., E)). We define the norm of x E Lpq to be the norm of { x k} in lq ( Z c, Lp ( IQ!k, E)) . Clearly, Lpp coincides with Lp. We list the implicit formulae of the norms on Lpq for

• This research was supported by grant No. 98-01-01035 from the RFBR Foundation. t Department of Applied Mathematics, Lipetsk State Technical University, 30,

Moskovskaya St., 398055 Lipetsk, Russia

323

324 V.G. KURBATOV

pf q:

llxll p < oo,

llxll q < oo,

llxll p,q < 00.

We note that in this definition one may change Qk to Qk. It is easy to see that Lpq are Banach spaces. For more information about the spaces Lpq, see [1, 4].

We denote by B = B(JB\, IF2 ) the space of all bounded linear operators acting from a Banach space lF1 to a Banach space IF2 ; if IF1 = lF2, we write briefly B = B(IFI). We denote by 1 E B(IFI) the identity operator. We denote by Nr = Nr(Lpq), 1 S r S oo, the set of all operators N E B(Lpq) possessing the following property. There exists a non-negative function {3 E Lr1 (JRc, JR) such that, for each x E Lpq and for almost all t E JRc,

(1) I(Nx)(t)l S j f3(t- s)lx(s)l ds.

IR'

In this case we say that the operator N is majorized by (the operator of) convolution with {3. Obviously, 11/311£1 S llf311Ld S 11!311£,1 S ll!31koo for 1 < r < s < oo. Therefore Nco C Ns C NrC N1 for 1 < r < s < oo.

Our aim is to prove the following theorem.

THEOREM 1.1. Assume that N E Nr(Lpq), 1 S r,p,q S oo, and the operator 1 + N is invertible. Then the inverse operator (1 + N)-1 can be represented in the form 1 + N1, where N1 E Nr(Lpq), too.

For the case r = oo, this theorem was first announced in [2]; see [3, theorem 5.3.6] for the proof. For r = 1 and a finite-dimensional JE, it was proved in [3, theorem 5.4.7]. Here we consider the case of arbitrary lE and r.

Also see [3, §6.4] for possible applications of assertions of the kind of theorem 1.1 to Green's function for functional differential equations.

2. Some Properties of the Class Nr· LEMMA 2.1. Let {3 E

L 1 (JRc, C) and x E Lpq(lRc, C). Then for almost all t E JRC, the function

SOME ALGEBRAS OF OPERATORS MAJORIZED BY A CONVOLUTION 325

s >--+ f3(t- s)x(s) belongs to L1 , and the convolution

(2) ({3 * x)(t) = j f3(t- s)x(s) ds

IR'

is a function of the class Lpq with 11!3 * xiiLp, :S: 2cllf31JL, · llxiiLp,.

Proof. The proof can be found, e.g., in [3, theorem 4.4.11]. D

COROLLARY 2.1. Let an operator N E N'r(Lpq) be majorized by the convolution with a function f3 E Lrl· Then

(3)

Proof. The proof easily follows from lemma 2.1 and formula (1). D

COROLLARY 2.2. The set N'r(Lpq), 1 :S: r :S: oo, is an algebra. The set N,(Lpq), r < s :S: oo, is a two-sided ideal in Nr(Lpq)·

Proof. The proof follows immediately from lemma 2.1. D

COROLLARY 2.3. Let n : IRe x IRe -+ B(JE) be a measurable function. Assume lln(t, s)ll :S: f3(t- s), where f3 E Lrl (IRe, IR). Then the formula

(4) (Nx)(t) = j n(t, s)x(s) ds

IR'

defines an operator N E Nr ( Lpq) majorized by f3.

See also theorem 2.7 below.

Proof. The proof follows from lemma 2.1 and Fubini's theorem. D

LEMMA 2.2. Let sequences Xn, Yn E Lpq(IRc, IR) converge in the norm to x, y E Lpq(lRc, IR), respectively. If Xn(t) :S: Yn(t) for almost all t, then x(t) :S: y(t) for almost all t, too.

Proof. We rewrite the inequality Xn(t) :S: Yn(t) as Yn(t)- Xn(t) ~ 0 and take a subsequence Yn, - Xn,, which converges almost everywhere. Passing to the pointwise limit in Yn, (t) - Xn, (t) ~ 0, we obtain x(t) :S: y(t). D

THEOREM 2.1. An operator N E N'r(Lpq), 1 :S: p, q :S: oo, possesses a natuml continuation to an operator N E N'r(Lpq) for any 1 :S: jj, ij :S: oo.

Proof. Let N be majorized by the convolution with a function f3 E £,.1 .

326 V.G. KURBATOV

First suppose that ij 'i oo. In this case Lpq n Lvii is dense in Lvii· From estimates (1) and (3) we conclude that N maps Lpq n Lfiii into Lvii and is bounded in the norm of Lvii· Hence it possesses the extension by continuity to the whole of Lfiii. By lemma 2.2, this extension satisfies (1).

Suppose ij = oo. Repeating the above argument, we may proceed from Lpq to Lpq and thus suppose without loss of generality that p = p. We take an arbitrary x E Lfiii. We set Xk = 1<tb. x, where 1Q. is the characteristic function

of the set ijk. Then we represent x as the series 2.: Xk (this series converges kE'llc

a.e.). Clearly, xk E Lpq· We consider the functions Yk = Nxk. For them from (1) we have IYk(t)l:::; (fh lxkl)(t) for almost all t. By lemma 2.1, the function fJ * lxl is defined a.e. and belongs to Lvii· From definition (2) of a convolution it follows that the series 2.: (fJ * lxkl)(t) converges to (fJ * lxl)(t) at each

kEZc

point t, where (fJ * lxl)(t) is finite. Therefore at the same t's the estimate 2.: IYk(t)l :::; 2.: (fJ * lxkl)(t) :::; ({J * lxl)(t) holds. This estimate shows

kEZc kEZc

that the series 2.: Yk(t) converges a.e. absolutely and defines a measurable kEZc

function y, and ly(t) I :::; (fJ * lxl)(t) for almost all t. By lemma 2.1, y E Lvii· :: We set Nx = y. 0

For N E Nr(Lpq), we denote by ON Or the infimum of llfJIIL"' over all fJ satisfying ( 1).

THEOREM 2.2. For any N E Nr, there exists the smallest non-negative fJ E Lr1 (1Rc,JR) satisfying (1). Obviously, ONOr = llfJIIL"'·

Proof. We fix N E N;. and denote by B the set of all fJ satisfying (1). Thus ON Or = inf{ llfJII£,1 : fJ E B }. Clearly, if fJ1, fJ2 E B, then min{ {J1, fJd E B, too. We choose a sequence flk E B such that llfJkll£, 1 < ONOr + 1/k. Replacing flk by min{fJI,fJ2, ... ,{Jk}, without loss of generality we may assume that flk+ 1 :::; fJk. In this case the sequence fJk converges pointwise to a function 13 E Lri· Clearly, 111311 :::; ONOr and 13 satisfies (1). Therefore 111311 = ONOr· Let fJ E B. It is easily seen that min{ {J, fJ} E B. If the set { t : fJ(t) < 13(t)} were not null, the Lr1-norm of the function min{ {J, fJ} E B would be less than ONOri but this contradicts the identity 111311 = ON Or· Hence 13 is the smallest element in B. 0

THEOREM 2.3. The algebra Nr = Nr(Lpq) is Banach with respect to " the norm 0 · Or·

Proof. From lemma 2.1 it follows that N >--t ONOr is a semi-norm. Assume that ON Or= 0. Then inequality (1) holds for the zero function fJ in

Lr1 , which implies that N is the zero operator. Thus 0 · Or is a norm. The completeness of Nr follows easily from 1.1.2. D

THEOREM 2.4. The set N"' 1 ::; r ::; oo, does not contain scalar operators a1, a E IC, a# 0.

Proof. Suppose the contrary: let N = 1 E Nr(Lpq)· Then, by theorem 2.1, N = 1 E Nr(L1) (note that in the proof of theorem 2.1 the extension N E Nr(Lpq) for p = ij = 1 is realized by continuity). We take the a-shaped sequence Xn(t) = e · 1[-1/n,l/n]'(t), where e E IE, llell = 1, is a fixed vector and 1[-l/n,l/n]' is the characteristic function of [-1/n, 1/n]c. By (1), I(Nxn)(t)l :':: ({3 * lxnl)(t), which tends to zero uniformly. But, on the other hand, lxn is Xn. These two conclusions can not be true simultaneously. D

THEOREM 2.5. The class Nco is dense inN"' r < oo, in the norm 0 · Dr· j.e. for arbitrary N E Nr and c > 0 there exists N E Nco such that ON- NOr< c.

Proof. As an auxiliary tool, for some bounded measurable functions a : Rc x Rc -+ IC, we would like to define the operation 9a : Nr -+ Nn which acts on an integral operator of form ( 4) as follows:

((QaN)x)(t) = J a(t, s)n(t, s)x(s) ds. JR:C

Unfortunately, we need this operation when N is not obligatorily an integral . operator; therefore the above direct definition may be meaningless.

First we suppose that a is simple, i.e. there exists n E N such that a takes the same common value on each subset (~ x ijm)/n, where k, mE zc. In such a case we set

((QaN)x)(t) = (N(a(t, ·)x))(t),

where a(t, ·)x means the functions>-+ a(t,s)x(s). Further we suppose that la(t, s)l :':: 1(t- s) for some bounded non-negative measurable function f.

Then the operator 9aN is well defined and, moreover,

(5) I(QaNx)(t)l :':: J 1(t- s){J(t- s)lx(s)l ds.

Ill'

Thus 9aN E Nr. In particular, by the definition of 0 · 0,

(6)

328 V.G. KURBATOV

Next let a be a function of the form a( t, s) = a( t- s)' where a : JE.C -+ c is a uniformly continuous bounded function. We take an arbitrary sequence an of simple functions, which converges uniformly to a. From estimate (6) it follows that 9anN is a Cauchy sequence in the norm U · Ur· So we can put 9aN = lim9anN. Clearly, this definition does not depend on the choice of

n the sequence an.

We show that for functions a of the considered kind

(7) I(QaNx)(t)l S j la(t- s)I,B(t- s)lx(s)l ds. IR'

Suppose lan(t, s) - a(t, s)l sen for all t and s, where en-+ 0. Then

From (5) and this estimate we obtain

I(QanNx)(t)l S J (la(t- s)l + en),B(t- s)lx(s)l ds. IR'

We briefly rewrite this inequality as un(t) s vn(t). By the proved, the sequence 9anN converges to 9aN in the norm U · Ur and hence, by corollary 2.1 and theorem 2.2, in the usual operator norm. Therefore the sequence Un = 9anNx converges in the norm of Lpq· On the other hand, from lemma 2.1 we conclude that the sequence Vn = ( (Ia I+ en),B) * lxl converges to Ia I* ,8 in the norm of Lpq· It remains to refer to lemma 2.2.

In a similar way one proves that

I((N- 9aN)x)(t)1 s j 11- a(t- s)I,B(t- s)lx(s)l ds. IR'

Then from the definition of 0 • 0 we obtain

(8) and

where 1- a is the function h H 1- a(h). Finally, let r < oo, and let a be a function of the form a(t, s) = a(t- s),

where a : JE.c -+ C belongs to L00 • Clearly, there exists a sequence an of continuous functions with compact supports such that II (an - a),BIIL" -t 0. From the first estimate in (8) it follows that 9anN is a Cauchy sequence in the norm 0 ·Or (here as usual an(t, s) = an(t-s)). So we can set 9aN = lim9anN.

n

It is easy to see that this definition does not depend on the choice of the sequence On·

Passing to the limit in (7) and (8) written for an, it is easy to see that estimates (7) and (8) remain valid for this class of functions a.

Now we come to proving that Noo is dense in Nn r < oo. Indeed, clearly, L 001 (JE.c, JR.) is dense in Lr1 (JE.c, JR.). Hence, for any E > 0,

we can choose fJ E Lool (JE.c, JR.) such that II.B- fJIILr 1 < E. Evidently, without loss of generality we may assume that 0 ::; {J(h) ::; ,B(h). We introduce a function ii as follows:

ii(h) = { ~ if ,B(h) =f. 0, 0 if ,B(h) = 0.

Clearly, {J(h) = ii(h),B(h). We set N = 9aN. Then by the second estimate in (8) we have ON- NOr= ON- 9aNOr ::; 11(1 ii),B~L,J ::; 11,8- fJI!L,j <E. On the other hand, from estimate (7) we obtain that N satisfies the estimate

I(Nx)(t)l = I(QaNx)(t)l::; J ii(t- s),B(t- s)lx(s)l ds ill"

- J f;(t- s)lx(s)l ds. ill"

Thus N E N00 • 0

We call any family {T;j} = {Tij E B(Lp(Qlj,lE),Lp(IQl;,JE)) i,j E zc} a matrix. For any matrix {T;1}, we set

ak = sup ll'lijll, k E zc i-j=k

This is the supremum norm of the k-th diagonal. We denote ~Y,

the set of all matrices for which the number

~{TijH = ~-~:;ak kEZc

is finite. We call a matrix of the class s a matrix with summable memory. Let us represent the spaces Lpq(lE.c,lE) as the spaces lq(Z<,Lp(Qk>JE)).

Assume that we have a matrix {Tij} E s. Clearly, the rule

(Tx)i = LTijXj, i E zc, jEZc

330 V.G. KURBATOV

induces an operator T E B(Lpq, Lpq)· Conversely, let JJ : Lp(Qj, lE) --+ Lpq(lR.c, JE) be the natural embedding,

and let Qi : Lpq(lR.c, JE) --+ Lfi(QJ, JE) be the natural projection. Then to each operator T E B(Lpq, Lfiq) we assign the matrix { Tij} defined by the rule

T;J = QiT JJ.

Thus we have obtained two operations {Tij} f-7 T and T f-7 {TiJ}· It is interesting to note that the matrix of an operator T can induce an operator different from T, see a counterexample in [3, example 1.6.4].

THEOREM 2.6. The class N00 (Lpq) coincides with the family of all operators induced by matrices of the class .s(zc, L1 (QJ, JE), Loo (Qi, JE)). In this case z-cHNiJH .s ONOoo .S zc~{NiJH·

This theorem shows that our definition of the class N 00 is equivalent to that in [3].

Proof. Let N E N'00 • From (1) for its matrix entries Nij we have

I(NiJx)(t)i .S j (J(t- s)lx(s)l ds, t E i + (0, 1]c,

j+(0,1]'

which implies

I(NijX)(t)i .S esssup (J(h) j lx(s)i ds. hEi-j+(-1,1)'

j+(O,l]'

From this estimate it is evident that Nij maps L1 (QJ,JE) to Loo(Q;,lE) with the norm

IIN;J : L1 --+ Looll .S ess sup (J(h) = L ess sup (J(h). hEi-j+( -1,1)' kEi-j+{ _1,0}' hE</&

Consequently, HNijH .s zcllfJIIL,=· Clearly, the matrix {N;j} induces the initial operator N.

Conversely, assume {N;J} E .s. Then we have

I(N;Jx)(t)l .S j ai-Jix(s)l ds,

j+(0,1]'

which implies

(9) I(Nx)(t)l .S L j ai-Jix(s)l ds, jEZ'j+(0,1]'

tEi+(0,1]",

t E i + (0, 1]<.

We set (3(h) = max ak. Clearly, (3(h) :::; I; ak. Thus llf311L,= :::; kEh+(-1,1)' kEh+(-1,1)'

2c H Nij }0. For such a (3, in formula ( 9) we have ai- j :::; max ak = (3(t- s). Hence (9) implies (1).

kEt-s+( -1,1)'

LEMMA 2.3. Let X be a Banach space, and let X 0 be its dense sub-oo

space. Then each x E X can be represented as x = I; xk, xk E X0, with k=1

00

I: lhll < oo. k=1

Proof. We take zk E X 0 such that llzk- xJJ :::; 2-k. Then we set x1 = z1, n

Xk = Zk- Zk-1 fork> 1. Clearly, I; Xk = Zn and JJxkJJ:::; 2-k + 2k-l. k=l

THEOREM 2. 7. Let the space lE be finite-dimensional. Then any operator N E Nr can be represented in form ( 4), where n : lRc x JRC -+ B(E) is a measurable function satisfying the estimate Jln(t, s)JJ:::; (3(t- s) with some (3 E Lr! (JRC, JR).

In particular, this theorem shows that our definition of the class N 1 is equivalent to that in [3].

Proof. It suffices to consider the case lE = C. Let N E N00 • We consider its matrix entry Nij for some i, j E zc. Let z E L 1 ( iji x ijj, IC) be a simple function, i.e. there exists n E N such

that z takes a common value on each subset (ijk x ijm)/n, where k, mE zc. We set

b(z) = J (Nijz(t, ·))(t) dt,

iji

where z(t, ·) is the functions 1-t z(t, s). It is easy to see that

(10) Jb(z)l :S: .I k,xQ; (3(t- s)Jz(t, s)J dtds.

Since N E Noo, the function (3 is bounded. Therefore the functional b is continuous in the norm of f 1. Qonsequently, b possesses an extension by continuity on the whole L1 («:Jli x «:JlJ,iCJ· Hence,

b(z) = ~-~ _ nij(t,s)z(t,s)dtds • }iji XQj

332 V.G. KURBATOV

for some function n;J E L00 (Q; x ijJ, C). We consider the operator

(N;Jx)(t) = j n;J(t, s)x(s) ds.

iQ.;

We observe that for simple functions y E L1 (Q;,CC) and x E L1(Qj,q (we say that a function x E £ 1 ( ijJ, C) is simple if ~ere exists n E N such that x takes the same common value on each subset (h,jn, k E zc)

j (N;Jx)(t)y(t) dt = j (N;Jx)(t)y(t) dt.

<()>. ij,

Since the subspace of simple functions is dense in £ 1 , from this identity it follows that N;J coincides with NiJ.

From (10) it follows easily that ln;J(t, s)l::; f3(t- s). Consequently, the whole operator N possesses representation (4) with ln(t, s)l::; f3(t- s).

Now let r < oo and N EN,. According to theorem 2.5 and lemma 2.3, 00

we represent N as a series N = :L Nk> Nk E Noo, absolutely convergent in k=l

the norm 0 · O,. By the proved, Nk has the form

(Nkx)(t) = j nk(t, s)x(s) ds,

Ill.'

00

where lnk(t,s)l::; f3k(t s) for some f3k E Lool with L:; 11/311£"'::; oo. Sum-k=I

ming up these equalities, we obtain the representation ( 4) for N. D

3. The Proof of the Main Theorem. Proof of theorem 1.1. We recall that for the case r = oo, the theorem is proved in [3, theorem 5.3.6].

We denote by Nr = Nr(Lpq) the algebra N,(Lpq) with an adjoint unit, i.e. the algebra of all operators a1 + N E B(Lpq), where N E Nr(Lpq(lRc, IE)) and a E C. We endow Nr with the norm Oa1+NOr = lal+ ONO,; according to theorem 2.4, this definition is unambiguous. From theorem 2.3 it is easy to see that N,. is a Banach algebra.

According to theorem 2.5 we choose N E Noo such that ON -NO,< 1/2. Then, since N, is a Banach algebra, the operator 1 + (N - N) is i~vertible inN, and (1 + (N- N)f1 = 1 + N1, where N1 E Nr· .


We consider the operator

K = (1 + (N- N)r1 (1 + N) = (1 + (N- N)r1 (1 + (N- N) + N) = 1 + (1 + (N- N))-1N.

We observe that the operator K is invertible (in B(Lpq)) as a product of two invertible operators (1 + (N- N))-1 and (1 + N). Since N E Noo and (1 + (N- N))-1 E Nr, by corollary 2.2 we have (1 + (N- N))- 1N E N00 •

Therefore (since for r = oo the theorem is known) K-1 E J\1;;, and K- 1 = 1+ K 1 , K 1 E N>O. Now from the representation (l+N)-1 = K- 1 (1+(N -N))-1

it is clear that (1 + N)- 1 = 1 + N1 , N 1 E Nr. D

REFERENCES

[1] J.J.F. Fournier and J. Stewart, Amalgams of LP and lq, Bull. Amer. Math. Soc., 13, 1 (1985), 1-21.

[2] V.G. Kurbatov, On algebras of difference and integral operators. Funktsional. Anal. i Prilozhen., 24, 2 (1990), 98-99.

[3] V.G. Kurbatov, Functional Differential Operators and Equations, K!uwer Academic Publishers, Dordrecht-Boston-London, 1999.

[4] J. Stewart and S. Watson, Which amalgams are convolution algebras? Proc. A mer. Math. Soc., 93, 41 (1985), 621-627.


VOLUME 8 2001, NO 3-4 PP. 335-344

MONODROMY FOR THE KZ EQUATIONS OF THE BwTYPE AND ACCOMPANYING ALGEBRAIC STRUCTURES *

V.P. LEXIN t

Introduction. A general method of the description for the monodromy of formal generalized Knizhnik-Zamolodchikov (KZ) equations associated with a root systems was proposed [3],[4]. Drinfeld [5] have obtained explicit formulae for a monodromy representation of the KZ equation of the An-type. He also described the quasibialgebra structure closely connected with this monodromy representation. Combinatorial and geometric interpretation of the quasibialgebra structure and Drinfeld's formulae was given [1], [2], [10], [11] and also the construction of a universal Kontsevich-Vassiliev invariant of knots and links using this quasi bilge bra structure was proposed. The axioms of the quasibialgebra structure can be interpreted as the condition of permutability of a holonomy of the KZ equation with coface maps of the co-simplicial set structure on the path space in the configuration one. The subject of this paper is the description of the quasibialgebra structure of the B-type. Permutability for the holonomy of the KZ equation of the B-type with coface maps of the B-type is proved. The B-analogue of the explicit Drinfeld formulae is obtained. In conclusion, the construction of the universal Kontsevich-Vassiliev invariant of the B-type for knots with additional symmetry of the order two is discussed.

1. Symmetrical Configuration Spaces. Let C~ = {(-zn, ... ,-z1,

0, z1, ... , Zn)lzi E IC} C IC2n+1 be the configuration space of ordered symmetrical collections of pairwise distinct points on complex plane. Let us define

' This work was partially supported by INTAS grant no. 97-1644. t Kolomenskiy Pedagogical Institute, KPI, Dept. Algebra and Geometry

335

336 V.P. LEXIN

the rrlaps ">: · c' n ~ d'm+l " 1 b the I u, . B , \0B , " = , ... , n y ru e

where z;+l = Zi + o, zj = Zj, j > i+ 1 and o is a positive real number less than the minimal distance from the point zi to other points Zj, j ~ i, j = 1, ... , n in IC. The maps~? will be also denoted by ~y(o). Let us introduce also the maps ~8, ~~+l that add a point z0 E IC very close to 0 and a point Zn+2 close to the infinity, respectively. Let the mappings cf : Cs -----? q-1

, i = 1, ... , n be defined by rules

where zj = zi+l> j = i+ 1, ... , n-1. We shall not distinguish two elements of C8 that differ only by a very small displacement of one point Zj, j = 1, ... , n along the real axis of the complex plane. Then the collection { C8, ~?, cf} forms a co-simplicial set, that is, the following axioms are fulfilled

~n+1 0 ~>:' = ~n+l 0 ~~ i < J. J t t J-ll

c~ o cn+l = c>:' o c~+l i < J. J 2 z J+l' - '

{

~~-1 o c'J_1, i < j c'J+1 o ~~ = idn, i = j or i = j + 1

A n-1 n · · + 1 '-'i_1 oci,t>J .

We consider the path space P(C'B) = hlr: [0, 1] -----? C'B}- The graph in lR x C of a path 1 = ( -zn(t), ... , -z1(t), 0, z1(t), ... , zn(t)) (tangle in Fig.1) have the.rotation symmetry of the order two around the dashed axis.

Fig. 1

MONODROMY FOR THE KZ EQUATIONS OF THE B,-TYPE 337

On {P(Ca)}n>l there exists a co-simplicial set structure {P(Ca), D.j, c:i} that is similar- to the co-simplicial structure {Ca, D.j, t:i}. Let pr(Ca) C P(Ca) be the subspace of paths that have real ends (i.e. in R'j,). Now we introduce the following list of elementary paths r~,; 1 ;2 ( t), r~,;( t), <p~.ili,is ( t), 'P~,i 1 i2 where the path with the subscript "-" is inverse to the path with subscript "+".

rA+ . . (t) = ,2122

I

r:"'i Fig. 2

"'y /!\

-Zi I Zi

THEOREM 1. An arbitrary path in pr(Ca), up to a homotopy with fixed ends, is a product of elementary paths.

Proof. The proof is similar to the proof of the presentation of an ordinary braid via generators.

The following examples of the decomposition into a path product will be needed:

338 V.P. LEXIN

Similar equations hold for the paths rA:,;1; 2 ( t), rB,i(t), 'PA:,i,;,i, ( t), 'PB,i,iz. Note that each map tl.f or ci is multiplicative with respect to the path

product in P(Cb). D

2. Algebra of Symmetrical Chord Diagrams. We consider the algebra P~(B) of symmetrical chord diagrams on 2n parallel lines. The algebra P~(B) is generated by the following chord diagrams tij, t1J, t~, (see Fig. 3):

t1; = J.+J -n -J -i '' ·-1 1 J

n

~- +---t-t+----1 ~ I 1 -n -J -1 J

n

~--H--n -i I 1 i n

Fig. 3

The generating relations are the following commutator equations:

1) [tij, t'jk + tjk] = [tij, tik + tjk] = [t;~, tij + tjk] = 0], i "# j "# k,

(2) 2) [tij + t? + tJ, t1;J = [t1; + t? + tJ, tij] = [tij + t1; + t?, tJ] = 0, i "# j,

MONODROMY FOR THE KZ EQUATIONS OF THE En-TYPE 339

3) [t;:;, t;tl = [t;:;, ttl = [t;:;, tZJ = [t~, t2J = o.

Sums and products of diagrams in the algebra P~(B) are defined as for ordinary chord diagrams (see [8], [9], [10]).

The multiplicative map

f:J,n,alg . ps(B) ----+ ps (B) t • n n+l

is defined according to the following rule: the i-th and ( -i)-th lines of the diagram D are duplicated and renumeration is done. The chords of the diagram D that end at the i-th and ( -i)-th lines are also doubled, and then only symmetrical resulting diagrams are left and summarized. The map

C:n,alg . ps (B) ----+ ps (B) t • n n-1

is defined according to the rule: if in the diagram D E P~(B) we have no chord ending at i-th and ( -i)-th lines, we simply remove them. If at least one chord ends at these lines, we set c:f(D) = 0. The set

is a co-simplicial set.

3. The KZ Equations of the B-type. A formal KZ equation associated with the root system of the En-type (KZn(B)) is an integrable Fuchsian-Pfaffian system on en of the following form: (3)

d\l!(z) = tst:1sn (tijdlog(z;- zJ) + t;jdlog(z; + zJ)) + ttfdlogz;} i!i(z),

where the coefficients t~, t? are formal constants satisfying commutator relations (2) from the previous section. It can be shown that these relations are equivalent to the integrability conditions [lKZn(B) 1\ [lKZ,(B) = 0 for the differential 1-form from equation (3)

n

nKZ,(B) = L (tijdlog(z;- Zj) + t;jdlog(z;. + Zj)) + 2:tfdlogz;

Equation (3) can be also considered as the equation for "horizontal section" of the flat formal K Zn(B) connection \7 KZ,(B) = d- nKZn(B)· Therefore we

340 V.P. LEXIN

can speak about a holonomy of equation (3). The holonomy along a path 'Y is defined by means of the Chen series

(4) HKZn(B)('Y) = 1 + J nKZn(B) + ... + f(nKZn(B))m + ... ,

where J(OKZn(B))m, m ~ 1 is them-iterated Chen integral. 'I

Solutions of the K Zn(B) equation are formal power series with non-commutative independent variables t'fj, t? and with analytic coefficients from commuting complex variables z1 , ... , Zn·

Further, we will consider the coeficients tij, tt, t? of KZn(B) as the chord diagram from section 2.

Equation (3) is invariant with respect to the action of the Weyl group W(Bn) = Sn · Z~ of the En-type root system that is the semidirect product of the permutation group Sn and direct product Z~ of groups order two. The action is defined by rules:

(5)

S • t'fj = f;_,(i)r'(j)' S E Sn, S • t? = t~-l(i)'

Ti · ttz = ti:z, i = k, l, Ti E Z~,

Ti · ttz = ttz, i of k,l,

S • (z!, ... , Zn) = (zs-'(1)' ... , Zs-I(n)),

Let P Bn(B) = 1r1 (Cs) be the Brieskorn pure braid group of B type. Using the isomorphism Bn(B) ""' PBn(B) · W(Bn), the representation

PnKzn(BJ : PBn(B) --......t P~(B) and the W(Bn)- invariance of the KZn(B) equation, it is possible to construct the monodromy representation

PKZn(B) : Bn(B) .......... x'k:oP~,k. W(Bn)

in the semidirect product of the completed symmetrical chord diagram algebra P~(B) = xk:0P~,k(B) and the Weyl group W(Bn)· Here P~,k(B) is the linear space of symmetrical chord diagrams with k symmetrical chords. We suppose that all generating symmetrical chord diagrams tij, tt, t? have one symmetrical chord.

The group Bn(B) has the representation r!,ii+l (t), r~,l (t).

Bn(B) = (bl> ... , bn-1> clbibi+lbi = bi+!bibi+1; bibj = bjb;, li- jl > 1;

The generators b; and c have the geometrical representatives r.7\,ii+! (t), r~, 1 (t) that were defined above. Let H K Zn(B) be a holonomy (parallel transport) of KZn(B) equation (3). Suppose that HKZ,(B)(r.7\,12) = R.7\1 E Pi(B); HKzn(B)(<p.:t,123) = !1?~ 1 E P3(B), Hxzn(B)(rll) = R~1 E P{(B), HKZ,(B)(<pt12 ) = !1?~ 1 E P{(B). Then analogues of equations (1) hold, where instead of~, r A, TB, <pA, <p8 we take=, RA, RB, !!?A, !!?B. That is,

(6)

These equations are the axioms of the quasi-bialgebra of the B-type. The elements RA, R 8 and !!?A, <!>8 are called R-matrices and the Drinfeld associators of type A and B, respectively.

THEOREM 2. The holonomy H('-y), 7 E pr(Cn) of KZn(B) equation

commutes with maps 6.;', that is,

(7)

In particular, the monodromy representations PnKz,cBl and PKZ,(B) commute with 6.f.

Proof. We will prove the permutability only for the representation PnKz,(B)" For PnKz,(Bl' the proof is the consequence of the previous asser

tion. Let 6.;'(5) : C;J -t C'B+1, i = 1, ... , n be the map defined above and 7 E pr(C'JJ). Let us denote n~Z,(B) = (6.;'(5))'rlKz,+I(B)· Then we have

= 1 + J nKZn+I(B) + · · · + J (OKZn+I(B))'n + · · · 6;'(5)(~) 6;'(5)(-r)

342 V.P. LEXIN

The 1-form nt-Zn(Ei) is the isomonodromic deformation of the 1-form

n

L (tk'1d1og(zk zt)+ttzdlog(zk+zt))+ L;tZdlogzk+ l:Sk<l:Sn; k,l;f:.i k=f;i

+ L ((tik + t;+1k)d1og(zi- zk) + (t~+J + t]+1k+J)d1og(zi + zk))+ k¢:i,i+I

We obtain

PnKzn+1<BJ (D.?(o)(r) = Pn'kzn(B) (r) =

= 1 + f D'i<:zn(B) + · · · + f (D'i<:zn(B))m + · · · =

' ' =1+ J Ll7'a19 (DKZn(B))+ ... + J(D.~·alg(DKZn(B)))m+ ... =

' ' = fl?'alg(1 + J QKZn(B) +. · · + J (QKZn(B))m + · · · =

' ' = fl?•alg(paKZn(B)(r)).

If n + 1-th point tends to infinity, we easily obtain the permutability of the map D.~+J with the monodromy representations. This completes the proof. 0

By theorem 2, we have the following explicit formulae for the monodromy representation PK Zn(B).

THEOREM 3. The monodromy representation PKZn(B) takes the following values on generators of Brieskorn group Bn(B)

,o PKZn(B)(c) = <P1J 1 (t!2 ,tj2,t~)e'tT1\h(t!2 ,tj2,t~),

where <PA, <Pa are Drinfeld associators of A-type and B-type.

Proof. Combining [3], [4], [8, chapter XIX], we obtain for n = 3

,-PKZn(B)(bz) = <PA:1(t!z, t23)e~sz<PA(t!2, t23),

,o

PKZn(B)(c) = <P1J 1 (t!2 ,tj2 ,t~)eh1wa(t!2 ,tj2 ,t~),

Applying i - 1 times the map Li~tg and n - i times b.~+ 1 , k = i + 2, ... n - 1 to the second formula and using the previous theorem, we obtain forb;, i > 2 the required formulae. D

4. The Kontsevich-Vassiliev Invariant for Knots with Supplementary Symmetry. Let us consider the algebra P~(B) :::J P~(B) of symmetrical chord diagrams supplemented with non-horisontal chord. We will consider the elements of path space pr(C'}j) as symmetrical tangles with symmetrical non-associative structure on the ends determined by means of the closeness of end points.

The monodromy representation PKZn(B) defines the elements RA, Rs, <P A, <P 8 . The correspondence Z 8 is defined by the rules

'P! 123 -t <P!1 E P;(B), rjg 1 -t Rjg1 E P{(B), , ,

± 1 E -t fo'

where v E Pf is the symmetrical closure of the Drinfeld associator, and g+, E- are symmetrical cap and cup (see [1] and [13]). ZB determines the multiplicative map of the set of symmetrical non-associative tangles to the algebra symmetrical chord diagrams. The following analogue of Bar-NatanLe-Murakami-Piunikhin theorem [1, 10, 11] holds.

THEOREM 4. The restriction of the map Z 8 on knots with supplementary symmetry of order two determine.s the symmetrical isotopy invariant of symmetrical framed knots.

The desired knots exist, for example, the knots 52 , 72 , 92 ,910 , etc from Convey list have the required symmetry.

344 V.P. LEXIN

REFERENCES

(1] D. Bar-Natan, Non-associative tangles, Geometric Topology, AMS/IP Stud. Adv. Math. Soc.2.1, AMS, Providence, 1997, 139-183.

(2] P. Cartier, Construction combinatoire des invariants de Vassiliev-Kontsevich des noeuds, C. R. Acad. Sci. Paris Ser. I Math., 316(1993), 1205-1210.

(3] LV. Cherednik, Monodromy Representations for Generalized Knizhnik-Zamolodchikov Equations and Heeke Algebras, Publ. RIMS, Kyoto Univ., 27 (1991), 711-726.

(4] C. De Concini and C. Procesi, Hyperplane Arrangements and Holonomy Equations, Selecta Math., New Series, 1, 3 (1995), 495-535.

(5] V.G.Drinfeld, On quasi triangular quasi-Hopf algebras and a group closely connected with Gal(Qj<Q,) Leningrad Math. J., 2 (1991), 829-860.

(6] T. tom Dieck, Knotentheorie und Wurzelsysteme, I, II, Mathematica Gottingesis, 21, 44 (1993).

(7] J.Frolich and C.King, Chern-Simons Theory and Knot Polynomials, Comm. Math. Phys., 126 (1989), 167-199.

(8] C.Kassel, Quantum Groups, Grad. Texts in Math., 155, Springer-Verlag, NewYork, 1995.

[9] C. Kassel and V. Turaev, Chord diagram invariants of tangles and graphs, Duke Math. J., 92 (1998), 497-552.

[10] T .Q. Le and J. Murakami, Representation of the category of tangles by Kontsevich's iterated integral, Comm. Math. Phys., 168 (1995), 535-562.

[11] S. Piunikhin, Combinatorial expression for universal Vassiliev link invariant, Comm. Math. Phys., 168 (1995), 1-22.

[12] N. Reshetikhin, Quasi triangular Hopf algebras and invariants of tangles, Leningrad Math. J., 1 (1990), 491-513.

[13] V. Golubeva and V. Leksin, On two types of representations of the braid group associated with the Knizhnik-Zamolodchikov equation of the Bn type, Journal of Dynamical and Control Systems, 5, 4(1999), 565-596.

VOLUME 8

2001, NO 3-4 PP. 345-352

ON A CLASS OF NONLINEAR DIRICHLET PROBLEMS WITH FIRST ORDER TERMS '

L. MOSCHINI, A. TESEI t AND S.I. POHOZAEV 1

Abstract. Using the method of integral identities, we prove nonexistence results for nonnegative solutions to a class of semilinear Dirichlet problems with nonlinear first order terms.

1. Introduction. In this paper we investigate nonexistence of nonnegative solutions to the semilinear Dirichlet problem:

(1.1) {

-6.11 =A ( h(x), 'Vu )g(u) + f(x, u)

u=O

inn

on an.

Here 0 <;; Rn is a connected bounded domain containing the origin with C 1•"

boundary an, A E R, and

n au ( h(x), 'Vu) = :L hi(x) ax ;

i=l t

the functions h : n -+ Rn' g : JR+ _, JR+ and f : n X JR+ -+ R are given (h =: (hl, ... , hn)).

In [4] problem (1.1) has been investigated under the following assumptions:

(i) g(u) = 1;

• Work partially supported through TMR Programme NPE No. FMRX-CT98-0201. t Dipartimento di Matematica "G. Castelnuovo", Universita di Roma ((La Sapienza'',

P.le A. Moro 5, I-00185 Roma, Italia 1 Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, 117966

Moscow, Russia

345

346 L. MOSCHINI, A. TESEI, AND S.I. POHOZAEV

(ii) f(x,u) = a(x)uq-! + b(x)u8-

1, where a,b: !1-t lR and q,s E (1,2*) (2* denotes the Sobolev critical exponent);

(iii) >. h(x) = 'VV;(x), the function V; : S1 -+ lR and its distributional derivatives being defined almost everywhere.

It turns out that a bounded weak solution to (1.1) exists (under suitable assumptions on the exponents q, s) as long as a, b E C(Q) and V; E C(Q) n W1·n(S1 ). On the other hand, if a, b E W1•00 (S1) and V; is smooth only outside some singular set of zero n-dimensional Lebesgue measure, solutions with a certain prescribed regularity need not exist. In the above case the linearity condition (i) allows for a variational formulation of problem (1.1); then weighted Sobolev spaces can be used as a natural tool to investigate the problem. The possible disappearance of solutions even for subcritical values of the exponents q, s (unlike the case when V; is constant) is due to the effect of singular coefficients in the first order term, which results in a loss of compactness of the problem (see [4] for details).

It is the purpose of this note to show that similar nonexistence results for (1.1) can result from nonlinear first order terms, even if the coefficient h(x) is smooth. For the sake of simplicity, we limit ourselves to prove this

[; assertion for the model problem:

(1.2) { -~u = >.(x, \lu )uP-2 + uq-! u=O

inn, on 8!1 ;

however, similar results can be shown to hold for more general choices of h, g and f in (1.1). As in [4], here we make of the method of integral identities introduced in [5]. We mention that similar nonexistence results due to nonlinear first order terms - yet only in the case of space dimension n = 1 -were proven in [1]-[3].

2. Results. Let us first recall some general ideas of the method of integral identities with reference to problem (1.1). In this respect, the following definition is expedient.

DEFINITION 2.1. By a strong solution to problem (1.1) we mean any nonnegative function u E C(Q) such that:

(i) its distributional derivatives up to the second order are defined almost everywhere in !1;

(ii) u satisfies almost everywhere in S1 the first equation in (1.1) and vanishes on 8!1.

Let u be a strong solution to problem (1.1). Following {5], let us multiply the first equation in (1.1) by [(h(x), \lu) +au], where a is any real constant.

ON A CLASS OF NONLINEAR DIRICHLET PROBLEMS 347

Integrating over rt and taking into account the boundary conditions on art, we obtain by formal calculations:

J [div h(x) _a] l\7ul2dx _ t j [ah; _!!}!_au J dx-2 . k axk OXk OX; n z, =1 n

-J [div h(x)F(x,u)- af(x,u)u]dx- t J Fx,(x,u)h;(x)dx+ n t=l n

+>- j g(u)l(h(x), \luWdx- a>. j div h(x)G(u)dx = n n

(2.1) = -~ J l\7ul2 (h(x), v(x))ds. &!1

Here v(x) denotes the outer normal defined at every point x E 80 and

u

F(x, u) :=.! f(x, t)dt , 0

u

G(u) := j g(t)tdt . 0

The proof of identity (2.1) is well-known ([5]; see also [6]), thus it is omitted.

It is natural to ask under which assumptions the calculations which lead to identity (2.1) can be given a sound meaning. Firstly, suitable regularity assumptions concerning h, g, f are needed (e.g., h E Cl, g continuous and f a Caratheodory function). Secondly, the strong solution u itself must be sufficiently smooth. The latter point requires a specific investigation, depending on the particular functions h, g and .f under consideration. It is convenient to put off temporarily this problem by making the following definition.

DEFINITION 2.2. A strong solution u to problem (1.1) is said to be admissible if: (i) every integral in equality (2.1) is finite; (ii) u satisfies equality (2.1) for any a E R

348 L. MOSCHINI, A. TESEI, AND S.I. POHOZAEV

The following nonexistence result concerning problem (1.1) is an immediate consequence of identity (2.1).

THEOREM 2.3. Suppose that

(Ho) (h(x), v(x)) 2: 0 for a. e. X E 8\1.

If there is a E lR such that the left-hand side of (2.1) is positive for u ¢ 0, then no admissible nontrivial solution of (1.1) exists.

We apply the above nonexistence result to problem (1.2). In this case equality (2.1) reads:

(2.2)

+.\ j i(x, V'uJIZuP-2dx- n;.\ j uPdx =

n n

= -~ J (x, v(x))iV'u l2ds.

&n

Observe that for problem (1.2) assumption (Ho) amounts to the assumption that n is star-shaped with respect to the origin.

The following result will be proved.

THEOREM 2.4. Let n be star-shaped with respect to the origin; assume that p 2: q > 2. Then there exists At > 0 (depending on p,q,n) such that for any .\ > At no admissible nontrivial solution to problem (1.2) exists.

It should be noticed that Theorem 2.4 implies nonexistence of solutions even in the subcritical case q E (2, 2*), unlike the case .\ = 0. We refer to this case as the case of strong convection. In the opposite case of weak convection - namely, when p < q - we can prove nonexistence only for q > 2' (as in the case.\= 0); in fact, the following holds.

THEOREM 2.5. Let n be star-shaped with respect to the origin; assume that q 2: p > 2, q > 2'. Then there exists A2 > 0 (depending on p, q, n) such that for any 0 < .\ < A2 no admissible nontrivial solution to problem (1.2) exists.

The proof of Theorem 2.5 is similar to that of Theorem 2.4, thus it will be omitted.

Having proved nonexistence of admissible solutions to problem ( 1. 2), it is natural to address nonexistence of strong solutions. For this purpose, we go back to investigating conditions under which a strong solution to (1.1) is admissible. A general result concerning problem (1.1) is the content of the following theorem.

THEOREM 2.6. Let the following assumptions be satisfied:

(2.3)

(i) f, Fx, are Camtheodory functions (i = 1, .. , n) ; (ii) lf(x, u)i:::; c(1 + iulq-l)

for a. e. X E it and any U E JR. ; n

(iii) I 2:: h;Ji',,(x,u)l:::; c(iul + iuiq) i=1

for a. e. x E it and any u E JR.; (iv) lg(u)l:::; cluiP- 2 for any u E JR.; {v) h, ~ E L 00 (0) (i,.i = 1, .. , n)

with p, q E (2, 2'). Then any strong solution u E H 2 (0) to problem (1.1) is admissible.

Clearly, assumption (2.3) is satisfied by h, g and .f as in problem (1.2); hence by Theorem 2.6 any strong solution u E H 2 (0) to problem (1.2) is admissible. Then Theorem 2.4 implies the following nonexistence result concerning strong solutions to problem (1.2).

THEOREM 2. 7. Let n be star·-shaped with respect to the origin; assume that 2* > p 2: q > 2 and.\ > A1 . Then any strong solution u E H 2 (fJ) to problem (1.2) is trivial.

3. Proofs. First we prove Theorem 2.4.

Proof of Theorem 2.4. We limit ourselves to consider the case 2 < q < p; if p = q, the proof is easier, thus it will be omitted.

Let u be any admissible solution to problem (1.2). Since by definition u is a strong solution to (1.2) and every integral in equality (2.2) is finite, it is easily seen that u E C(n) n Hd(fJ ).

Due to Young inequality, there holds (pointwise in fJ):

Cl' 1 uq < -u2 + --1LP

- 11 vCv ' (3.1)

where C is anv positive constant, " := ~, I/ := l!:::c.'c22

• Let ~ r p-q q-

(3.2) n - > ()!" q- '

350 L. MOSCHINI, A. TESEI, AND S.L POHOZAEV

multipling inequality (3.1) by (a- ~) and integrating over n gives:

On the other hand, assuming

(3.3) n-2 -->a 2 - '

by Poincare inequality we obtain:

(n- 2 ) J 2 (n- 2 ) J 2 -2

- -a IY'ul dx 2: Ao -2

- -a u dx,

n n

where Ao denotes the first eigenvalue of -tl on n with homogeneous Dirichlet conditions.

Due to the previous inequalities, the left-hand side of (2.2) is greater than or equal to the following quantity:

(3.4)

+A jl(x,Y'uWuP-2dxn

Let there exist a E JR'., C > 0 and A E lR'. such that (3.4) is strictly positive for any nonnegative u E C(O), u ¢ 0; then by Theorem 2.3 no admissible nontrivial solution of (1.2) exists. Clearly, this is the case if the following additional inequalities (besides (3.2)-(3.3) ):

(3.5) (n- 2 ) C~' (n ) Ao -2

- - a - ---;; q -a 2: 0 '

(3.6)

(3.7) -+-· --IY <0 naA 1 (n ) p vcv q - '

are satisfied, at least one of them being strict; observe that inequalities (3.2), (3.6) and (3.7) necessarily imply a < 0. Rewriting (3.5) and (3.7) in the equivalent form:

cv > _P- (~ -· a) - vlalnA q '

C1, , (n- 2 ) :S; f.lAO -2- - IY

we obtain the following compatibility condition:

Since ~ = v- 1, condition (3.8) gives the following restriction on A:

A > _P_ (-v-)Ao :::..=__ -a ~ -a [ (

2 )]1-v( )v vnlal z; - 1 2 q

where a E ( -oo, min{O, n22 }). As an elementary calculation shows, the best choice of a in the above interval leads to the condition A> A1 , where:

A1 := (~) (~ ) f':i (-P ) (P- q) f':i > 0 . n AO p- 2 p- 2

Since a < 0, inequality (3.2) is obviously satisfied. On the other hand, if the compatibility condition (3.8) is satisfied, we can find C > 0 such that (3.7) and (3.7) are satisfied. Then by Theorem 2.3 the conclusion follows. D

Pr-oof of Theor-em 2.6. Let u be any strong solution to problem (1.1). It is sufficient to prove that under the present assumptions every integral in equality (2.1) is finite. In fact, it is easily checked that in such case the formal proof of identity (2.1) has a sound meaning; then the conclusion follows.

The following statements are immediately seen to hold:

(i) J ldiv h(x)II'Vu l2dx < oo, t J I ~hi 0°0 :~~ dx < oo . VXk Xk uX,

n z,k=ln

ifu E H 1 (<;l) (due to assumption (2.3)-(v));

(ii) .! ldiv h(x)IIF(x, u)ldx < oo, t .I IF-;,,(x, u)llh;(x)ldx < oo, n z=l n

./ lf(x, u)uldx < oo

n

352 L. MOSCHINI, A. TESEI, AND S.L POHOZAEV

if u E Lq(n) (due to assumptions (2.3)-(ii), (2.3)-(v) and to the definition of F);

(iii) j ldiv h(x)IIG(u)idx < oo,

0 if u E IJ'(r2) (due to assumptions (2.3)-(iv), (2.3)-(v) and to the definition of G);

(iv) j i'Vu l2 l(h(x), v(x))!ds < oo

iJO

ifu E H 1(8r2) (due to assumption (2.3)-(v)). Moreover, since any strong solution to problem (1.1) satisfies the first equation in (1.1) almost everywhere inn, there holds:

J!g(u)!l(h(x), 'VuWdx :S J!t:.u!l(h(x), 'VuWdx+ j!f(x,u)l!(h(x), 'Vu)!dx. n n n

It follows easily that:

(v) j !g(u)ll(h(x), 'VuWdx < oo

n ifu E H 2 (r2) (due to assumption (2.3)-(v)).

Since by assumption u E C(Q) nH2 (r2), every integral considered in (i) (v) above is finite (concerning (iv), observe that u E H 2 (r2) implies I'Vu I E H~(an) <:;; L2 (8r2) by embedding results). Then the conclusion follows.D

REFERENCES

[1] T. Chen, H. Levine and P. Sacks, Analysis of a convective Reaction-diffusion equation, Nonl. Anal., 12 (1988), 1349-1370.

[2] H.A. Levine, L.E. Payne, P.E. Sacks and B. Straughan, Analysis of a convective reaction-diffusion equation II, SIAM J. Math. Anal., 20 (1989), 133-147.

[3] S. Claudi, L.A. Peletier and A. Tesei, A nonlinear diffusion equation involving convection and singular absorption, Nota Scientifica 99/14 Dip. Mat. "G. Castelnuovo'', Universiti:t di Roma "La Sapienza", 1999.

[4] L. Moschini, S. I. Pohozaev and A. Tesei, Existence and nonexistence of solutions of nonlinear dirichlet problems , Nota Scientifica 99/32 Dip. Mat. "G.Castelnuovo", Universita di Roma "La Sapienza", 1999.

[5] S. I. Pohozaev, Eigenfunctions of the equation ilu+>.f(u) = 0, Soviet. Math. Dokl., 6 (1965), 1408-1411.

[6] P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.


VOLUME 8 2001, NO 3-4 PP. 353·-363

FOURIER-BESSEL TRANSFORMATION OF COMPACTLY SUPPORTED NON-NEGATIVE FUNCTIONS AND

ESTIMATES OF SOLUTIONS OF SINGULAR DIFFERENTIAL EQUATIONS*

A.B. MURAVNIK t

Abstract. This paper is devoted to the investigation of pure Fourier-Bessel transformation (see [2]) for non-negative compactly supported f:

00

- dcf 1 I v+! 1 j(ry) = ry" Y .fv(rJy)j(y)dy; V > -2. 0

Using the method proposed in [8] we strengthen the estimates obtained in [6] for arbitrary non-negative functions. Further those estimates are applied to differential equations with a singular Bessel operator, and we obtain the corresponding estimates for solutions of the mentioned equations: weighted £=-norms of the solutions are estimated from above by their weighted £ 2-norrns.

Introduction. a E (0,1!+ ~)

(1)

It is proved in [6) that iff is non-negative, then for any

(I' - 0:-1 -

llr 2 flloo ~ Cllr-, fll2,

where C depends only on 1/. This result is a certain generalization of a wellknown result of F. Mattila for the classical case of Fourier transformation (see [5)). In [8] P.Sjolin proved that the additional (besides the nonnegativity) assumptions of the compactness of the support and of the radiality improves the result of [5] in the following way: the weights in the right-hand side and in

' The author is very grateful to Prof. P.Mattila and to Prof. P.Sj6lin for attentive concern and fruitful discussions.

i" Moscow State Aviation Institutn, Rnssia J 25871, :\'loscm.v 1 Volokolamskoe shosse 4

353

354 A.B. MURAVNIK

the left-hand side are connected now by an inequality (instead of an equality in [5]) and the powers of weights belong to a wider interval than in [5].

In this work the method of [8] is applied to Fourier-Bessel transformation and gives the following estimates for non-negative f supported in [0, R]:

(2) f!.. ~ a:-1 -

llr 2 flloo :':: Cllr-, Jll2 ·

The main results are the following:

if a E (0, 2v + 1], then (2) holds for fJ E (0, a] and does not hold for fJ > a;

if a E (2v + 1, 2v + 2), then (2) holds for fJ E (0,2v + 1] and does not hold for fJ > 2v + 1.

The constant in (2) generally depends on v, a, (J, R. Further we use the above results to derive estimates of solutions of the

following singular equation:

(3) P(-B)u = J(y),

h B defB del 1 d ( kdu) . h B l k 2 1 · · · w ere u= ku= yk dy y dy IS t e esse operator; = v+ 1s a pos1t1ve

parameter; P is a polynomial with real coefficients. More exactly, under the assumption of non-negativity and weighted

summability of 1f~1), the following estimate for solutions of (3) is valid:

(4)

where suppj c [0, R], a E (0, k+ 1), fJ E (0, min(a, k)], C depends on a, (J, k in case of a :<:; k and depends only on a, k otherwise. For fJ > min( a, k), estimate ( 4) is not valid.

1. Preliminaries. In this section the notations and definitions, which are used. throughout the whole paper, are introduced; we also recall the necessary properties of Fourier-Bessel transformation.

00

Lp,k(O,+oo)~{f 111!11 = (j Yklf(y)[Pdy)~ < oo} for a finite p;

0

Loo,k(O,+ooJ~{JIIIfll =vraisupykiJ(y)l < oo}. The set of infinitely smooth functions with compact supports is denoted

by CQ'(R). The subset of C0 (R) formed by even functions is considered; the set of restrictions of elements of that subset to (0, +oo) is denoted by Co:'even(O, +oo). This Co:'even(O, +oo) will be the space of test functions.

FOURIER-BESSEL TRANSFORMATION AND SOLUTIONS' ESTIMATES 355

Distributions on CIJ':'even(O, +oo) are introduced (following, for instance, [1]) with respect to the degenerate measure ykdy:

00

(5) for any 'P E cr,;,ven(O, +oo) (!, <p)'i§f J yk f(y)<p(y)dy. 0

Thus all linear continuous functionals on C<\even(O, +oo), which can be given by (5) (with f E L 1,k,loc(O, +oo)), are called regular (and the corresponding function f is called ordinary).

The Fourier-Bessel transformation is introduced following [2]: 00

- dcf J k J(ry) ~ Y .iv(ryy)j(y)dy, 0

where .iv(z) is the normalized (in the uniform sense) Bessel function: . ( ) _ Jv(z)

Jv Z ---. zv 00

Note that f(y) =C.! rykjv(7!y)}(ry)dry. The generalized shift operator

0 corresponding to the considered degenerative measure is found in [4]:

n

r:j(y)~C J J(Jy2 +h2 -2yhcosB)sink-10dB 0

00 00

such that.! rykg(ry)TJ j(y)dry = .! ryk f(ry)TJg(y)dry and therefore one can in-

o 0 troduce the generalized convolution:

00

(f * g)(y)~.! ryk j(ry)TJg(y)dry such that f * g = jg (see also [3]).

0

In Fourier-Bessel images the Bessel operator acts as a multiplier: ~ = -ry2u.

2. The Case of the Power of the Weight not Exceeding the Parameter at the Singularity. Let k > 0, a ::; k. Hereafter all the absolute constants generally depend on v, a. Let a non-negative function f E

1

L1,k(O, +oo)nL2,k(O, +oo) and suppf c [0, 1]. Then ](r) = .fskjv(rs)f(s)ds. 0

356 A.B. MURAVNIK

1kaf3 C C Take {3 < a. v + -

2 = -

2 ;c: -

2 > -

2 =? ljv(t)l ::; - 1 ::; ...,- on [1, +oo).

t"+2 t\i On the other hand, jv(O) = 1. Hence there exists C such that jv(O) ::; -jk- on

t>

(0, 1]. Thus )v(t) ::; cd on (0, +oo). It implies:

00 00

= cr-~ j sk-;1 i(s)d8 = Cflr-~ j sk8~-k- 1 /(8)d8 = 0 0

1 00

= Cflr-~ j s~- 1 /(8)d8 + Cflr-* j 8~- 1 /(s)ds.

0 1

1 1 00

I J s~- 1 /(s)dsl ::; llflloo J 8~- 1 d8 = c(3 J yk f(y)dy because f ;c: 0. So it is 0 0 0

equal to

ex; 1 00 1 1 1 1

c13 ( j yk J(y)dy )' ( j xk !(x)dx )' = c13 ( j j xV J(x)f(y)dxdy )'. 0 0 0 0

Let y E [0, 1], hE [0, 1]; then

~

Th -a > C J sink-! ()d() d:;j C > C d~ ]__ y y - (y + h)" (y + h)" - 2" c

0

It means that

1 1 1 1

j f xkyk f(x)f(y)dxdy::; C j xk f(x) j yk f(y)T:y-"dydx =

0 0 0 0

00

= c(!, J ~ ) = c j y"-1 f2(y)dy d~ cr"(f).

0


00

Thus j](r)j::; C!lr-~ (I,(j)r~ + Cr-% J s%-1 ](s)ds ==?

1

00

JZ(r)::; Cllr-!3J,(f) +Cilr-13(j /-~-' s"2' ](s)dsr::;

1

00 00

::; C13r-!l I,(!)+ C13r- 13 I sf3-a-lds I s"- 1 JZ(s)ds = C13 r-!3 !,(!).

1 1

Therefore the following statement is proved:

THEOREM 1. Let k > 0, a E (0, k]. Then for any (3 E (0, a) there exists C!3 such that for any non-negative f E L1,k(O, +oo) n L2,k(O, +oo) with suppf C [0, 1]

(6)

00

JZ(r)::; C13r-!3 j y"-1 f2(y)dy for any r > 0.

0

l def

Now let suppf c [0, R, R > 1. Define fn(Y) = f(Ry); then suppfn C

[ 0, 1]. Hence Theorem 1 is valid for fn (y): for any (3 from ( 0, a) there exists C 13 such that for any r > 0

00

/n2(r)::; C13r-f3 I y"-1 fn

2(y)dy.

0

On the other hand,

00 00

fn(Y) =I skjv(ys)fn(s)ds = R:+l I skjv(~s)f(s)ds = R:+J(~y). 0 0

So

00 00

f2 G) ::; C!lr-!3 j y"- 1J2( ~y)dy = C!lR"r-!l j y"- 1J2(y)dy 0 0

358 A.B. MURAVNIK

00

for any R > 1 and any r > 0. That is, j2(r) :S: Cf3R"-f3r-f3 j y"-1j2(y)dy

0 for any positive r.

Thus the following statement is proved:

COROLLARY 1. Let k > 0, o: E (0, k]. Then for any {3 E (0, a) there exists Cf3 such that for any non-negative compactly supported f E

L1,k(O, +oo) n L2,k(O, +oo)

00

(7) P(r) :S: Cf3R"-f3r-f3 j y"'-1]2(y)dy for any r > 0

0

(8)

where R is the right-hand boundary of suppf.

This section is completed by the statement, which shows sharpness (in a certain sense) of the obtained results:

COROLLARY 2. Inequality {6) (and therefore (7) and {8)) is not valid for /3 > o:.

Proof. Suppose to the contrary that (6) is valid for some /3 > o:. Let f E L 1,k(O, +oo) n L2,k(O, +oo) be non-negative and suppf c [0, 1]. Let us

define fR(Y) d~ f(Ry) for R > 1; then suppfR c [0, *] c [0, 1] so (6) is valid for JR(y). Then, as in Corollary 1, P(r) :S: Cf3Ra-(3r-(3Ia(J) for any positive r; hence ]2(1) :S: Cf3R"-f3 I a (f) for any R > 1. It means (since a < /3) that }2(1) = 0.

We arrive at a contradiction. D

3. The Case of the Power of the Weight Exceeding the Parameter at the Singularity. THEOREM 2. Let k > 0, o: E (k, k + 1).

Then for any /3 E (0, k] there exists Cf3 such that for any non-negative f E L1,k(O, +oo) n L2,k(O, +oo) with suppf c [0, 1]

}2(r) :S: C(Jr-(Jla(J) for any r > 0.

k 1 1 /3 c Proof. v = 2 - 2 =?- v + 2 2: 2 =;- Jjv(t)l :S: t~ on [1, +oo). On the

c other hand, Jv(O) = 1 =?- there exists C such that Jjv(t)i :S: ~ on (0, +oo).

t2

Then we can repeat the proof of Theorem 1 completely, because 00 J sf3-a- 1ds < oo. 0

1 And, as above, we have the following corollary:

COROLLARY 3. Let k > 0, a E (k,k + 1). Then for any fJ E (O,k] there exists C13 such that for any non-negative compactly supported f E

L1,k(O, +oo) n L2,k(O, +oo)

00

j2(r):::; C13R"'- 13 r-!3 J y"'-1 j2(y)dy for any r > 0

0

where R is the right-hand boundary of supp f.

Sharpness (in a certain sense) of the results of this section is established by the following theorem:

THEOREM 3. Theorem 2 is not valid for fJ > k.

Proof. Let f(s) d;j fR(s) d;j e-iRs<p(s), where <p E C00 (R), supp<p C

(0, 1), <p?: 0, <p = 1 on ~~' t). Then

00 00

](R) =.! skjv(Rs)f(s)ds =.! hv(Rs)e-iRs<p(s)ds.

0 0

Taking into account that

C m-1 C 1 1

j (t) = -(~ - 1 cos [t _"!_(v -l + -)] + o(-)) = v t"+! L..... tl 2 2 tm

1=0

+0(--i-.) as t-+ oo, n+l

360 A.B. MURAVNIK

we obtain:

def ·1rk def ·1rk where C1 = Ce-'' , C2 = Ce'' . Observe that A is the coefficient at R-~. We have to prove that lim A # 0.

R->oo

1 1

A 1r;·, 1rj' C =cos 4k s'<p(s)ds +cos '4k s'<p(s) cos2Rsds+ 0 0

1 1

+sin~k j s~<p(s)sin2Rsds+i( -sin~k j s~<p(s)ds+ 0 0

1 1

+sin~k j s~<p(s)cos2Rsds+cos~k j s~<p(s)sin2Rsds). 0 0

Suppose to the contrary that lim ~A= lim \SA = 0.' Then lim (~A cos ~ k-R-+oo R-+oo R-+oo

\SA sin ~k) vanishes, too. But the last limit is equal to

1 1 1

Jk 1rfk 1r!k s' <p(s )ds +cos 2'k s' <p(s) cos 2Rsds +sin 2k s'i <p(s) sin 2Rsds 2: 0 0 0

(note that the integrated function is non-negative)

7 7

J.k 1r !", 1r 2: s'<p(s)[l + cos(2k- 2Rs)]ds = s'[l + cos(2k- 2Rs)]ds 2:

~ i

7

> (~) ~ j[l + cos(~k- 2Rs)]ds = (~) ~ G- sin(

i

- 2Rs) ~~) 2R 2

8

We obtain a contradiction.

R-too 1 (5) ~ O --+-- >. 4 8

Thus A does not tend to zero as R -+ oq. Moreover, there exists C > 0 (e.g. ~ ( ~) 2 ) such that for R large enough

~A cos ik- <;SA sin ik?: C.

Therefore, C ::; [~A[ + [S'A[ * C2 ::; (l~A[ + [S'A[)2 ::; 2[A[ 2

. That is, def /c ~ c k [A[ ?: Co = y 2 for R large enough. Hence lfn(R)l ?: "'fR-, for R large

enough. On the other hand, the following statement is true:

LEMMA 1. [fR(R)IZ ::; Cf!R-f! for any positive R.

Proof. Introduce f 1(s); (l = 1,4) as follows:

ft(x) d;j { ~f(x), if_~f(x) > 0. 0, otherwise '

fa(x) d;j { S'j(x), if _S'f(x) > 0 . 0, otherw1se '

fz(x) d;j { 0, if ~f(x) > 0 -~J(x), otherwise

def { 0, if S'f(x) > 0 !4(x) = -S'f(x), otherwise ·

So 0::; !1(x)::; lf(x)[, l = 1,4; f(s) = ft(s)- f2(s) + i[h(s)- !4(s)]. Then 4 4

l](sW::; CLft2(s)::; Cf!s-f!Lla(!ll due to Theorem 2.

l=l l=l Now observe that

for any l = 1, 4.

00 00

= C j j xVlf(x)llf(y)[T;y--"dxdy?: 0 0

00 00

C J .I xkyk fl(x)!l(y)T;y-"dxdy = Cla(Jl) 0 0

362 A.B. MURAVNIK

Thus IJR(rW::; Cflr-fl la(IJR!l· But IJRI does not depend on R. Lemma 1 is proved. 0

Thus there exists Cfl such that for R large enough

R-k ::; CfJR-fl.

It means that k 2': (J. Theorem 3 is proved. 0

REMARK. In the case k < a < k + 1 one can remove the dependence of Con (J.

In fact, that dependence arises (see the prove of Theorem 1) in two points:

1 00

Gil j s~- 1ds where .;::1 = Cg_s~-k- 1 and 2 2 J (j-a-1d 8 s.

0 1

r(k-li-1) c = -~- 2k-~- 1c

~ r(~) k (see for instance [7], pp.157-158)

1

=? C~ j s~- 1ds = ~C% is bounded by a constant depending only on k

0

(because r( ~) has a simple pole at the origin). On the other hand, co

J sfJ-a-1ds = ~(J ::; ~k does not depend on (J. (]!- 0!-

1

Note that the dependence of C on a cannot be removed (at least in that r( k-a-1)

way), because Ca = r(~) , and hence C becomes infinite at both ends of

(k,k + 1). Note also that for the case 0 < a ::; k one even cannot remove the

co

dependence of Con (J, because j sfl-a-1ds = a~ (J ---+ oo as (J ->a.

1

4. Estimates of Solutions of Singular Equations. In this section we apply the above results to estimate the norms of solutions of (3). Let u E L2,k(O, +oo) satisfy (3) at least in the sense of distributions. Then u also belongs to L2,k(O, +oo) (see [2]) and

(9)


If P(TJ2) E Lz,k,loc(O, +oo), u(rJ) E Lz,k,loc(O, +oo), then ](TJ) E Ll,k,loc(O, +oo),

that is, J(TJ) is an ordinary function. Thus (9) is an equality of ordinary functions and hence the following division is legitimate:

- ](TJ) u(rJ) = P(TJ2 ) E Lz,k(O,+oo).

Now we denote 1(~]) by g(rJ) and assume that g is non-negative and belongs

to L1,k(O, +oo); we also suppose that suppj C [0, R]. Then g satisfies the conditions of Theorem 1 - Theorem 2 and u = §. This implies the following statement:

THEOREM 4. Suppose that a E (0, k + 1). Let 1~1) be non-negative

and belong to L1,k(O, +oo), suppj c [0, R], and let u E Lz,k(O, +oo) satisfies equation (3) (at least in the sense of distributions). Then for any (3::; min( a, k) there exists C = C(a, (3, k) such that (4) is valid. Moreover, C does not depend on (3 in case of a> k.

REMARK. In the same way Corollary 2, and Theorem 3 imply that Theorem 4 is not valid for (3 > min( a, k).

REMARK. Note that in case of a= (3 ::; ~ the constant C depends only on k and the condition of the compactness of suppj is taken off (see [6]).

REFERENCES

[1] V. V. Katrahov, On the theory of partial differential equations with singular coefficients, Sov. Math. Dokl., 15, 5 (1974), 1230-1234.

[2] I. A. Kiprijanov, Fourier-Bessel transforms and imbedding theorems for weight classes, Proc. Steklov Inst. Math., 89 (1967), 149-246.

[3] I. A. Kiprijanov and A. A. Kulikov, The Paley-Wiener-Schwartz theorem for the Fourier-Bessel transform, Sov. Math. Dokl., 37, 1 (1988), 13-17.

[4] B. M. Levitan, Expansions in Fourier series and integrals with Bessel functions, Uspehi Mat. Nauk, 6, 2 (1951), 102-143.

[5] P. Mattila, Spherical averages of Fourier transforms of measures with finit.e energy; dimensions of intersections and distance sets, Mathematika, 34 (1987), 207-228.

[6] A.B. Muravnik, On weighted uniform estimates for pure Fourier-Bessel transform of nonnegative functions, Reports of Mittag-Leffler Inst., 33, 1994/1995.

[7] G. E. Shilov, Generalized functions and partial differential equations, Gordon and Breach, 1968.

[8] P. Sjolin, Estimates of averages of Fourier transforms of measures with finite energy, Ann. Acad. Sci. Penn. Math., 22, 1 (1997), 227-236.

. FUNCTIONAL DIFFERENTIAL EQUATIONS

VOLUME 8 2001, NO 3-4 PP. 365-369

BOUNDARY VALUE PROBLEM FOR ALGEBRAIC POLYNOMIALS

S. NIKOLSKH '

Let Rn 3 x = (x1 , ... , Xn) be then-dimensional Euclidean space and

P = PN(x) = L CaX"'

lai:SN

n

(N=0,1, ... )

algebraic polynomials of degree N defined on JRn. Here ea can be zero for lal = N, too. The trace of P = PN on the unit sphere C5 c JRn : PI,. is called a spherical function of order N.

In the theory of spherical functions there is a fundamental theorem that says: for every algebraic polynomial U = UN of degree N coinciding with P on e5: Ul,. = PI,.. One can find this theorem in text-book [1] by S.Sobolev in JR3 and in JRn in book [2] by J .Stein and G. Weis.

We generalize this theorem on a differential elliptic operator of the order 2l.

LU = L aa(3U(a+f3), lal=lfil=l

with constant coefficients.

aa(3 = a(3oo l = 1, 2, ...

We consider also algebraic surfaces C5 = C52t ( s = 1, 2, ... ) of degree 2s:

H(x) = 1, H(x) = L b"'13xa+f3 2 c L x2"', bafi = bfia·

l<>l=lfil=s lal=s

'Steklov Mathematical Institute, RAS, 117966 Russia, Moscow, ul. Vavilova, 42

365

366 S. NIKOLSKII

In the case s = 1 the surface fJ = fJ1 , is an elastica! ellipsoid in JRn, but for s > 1 it is only similar to an ellipsoid.

The inequalities 0:::; H(x) < 1 define an open set Q c JRn with boundary an= (J.

(1)

We consider the boundary value problem

LU=O, u<aJ I = p<aJ I

<7 "'

X E Q,

lal :<::: l - 1

for any given polynomial P = PN of degree N. It is well known that this problem has a unique solution U differen

tiable 21 times. But the question is whether U is an algebraic polynomial of degree N. We prove

THEOREM. 1) If s = 1, i.e. fJ = fJ1 is an ellipsoid, then for any polynomial P = PN of degree N the solution U of boundary value problem (1) is an algebraic polynomial of degree N, too.

2) But if s > 1, it is not so. There are polynomials P = PN, for which the solution U of problem (1) is not a polynomial of degree N.

Proof. We prove this theorem by Variational method. The methods witch were applied in the mentioned works [1], [2] are not suitable in our case.

Let us consider the bilinear form

E(.;a, 7!{3) = L aa{J.;a7J{h !al=lfJI=l

and the quadratic form

E(.;a) = E(.;a, .;/3) 2 C L 6a, !al=l

where .;,, 7Jf3 are variables depending on multiindices a, (3 and the inequality expresses ellipticity of the form and of the operator L.

For any given polynomial P = PN, we consider the set 9.np of polynomials <p = 'PN of order N, for which

There is an assertion: for any given polynomial P = PN there is a unique polynomial U = UN E 9Jtp, for which the following minimum principle holds:

(2) min £(<p) = £(v), <pE!!Jlp

BOUNDARY VALUE PROBLEM FOR ALGEBRAIC POLYNOMIALS 367

One can prove this assertion as they do in general case, when consider such a minimum in the Sobolev class WJ(S1) of functions with boundary values

(see [3]). It leads to a unique function Q(x) E WJ(S1), Q(")lu = p(a)!u

Ia! ::; l- 1 that automaticaly solves problem (1) (LQ = 0). In the given case we start from polynomials <.p = 'PN E !JJtp and as a

result obtain a polynomial U =UN E !JJtp of degree N. But now U can or can't be a solution of boundary value problem (1).

Note that the class 0010 consists of all polynomials v = v N of dergee N which satisfy zero boundary properties:

v(")l = 0 u ' Ia! ::0 l - 1.

Every polynomial <.p = 'PN E !JJtp can be represented as a sum <.p = U +v, for all v E 0010 and it is possible to write assertion (2) in an equivalent form:

and since

then in form

or

(2')

or

(2")

E(U + v) ~ E(U), for all v E 0010 , U E !JJtp

E(U + v) = £(v) + 2£(U, v) + E(v),

2£(U,v)+£(v) ~'

E(U,v)=O,

j (LU)v dx = 0, n

for all v E 0010, U E !JJtp

for all v E 001o, U E !JJtp.

368 S. NIKOLSKII

We have obtained equivalent relations:

(2) ;::::t (2') ;::::t (2"),

from which it will be used

(3) (2) ;::::t (2").

Thus if U E 9Jl:p satisfies (2), then it satisfies also (2") and conversely. Note that any polynomial v = VN E 9Jl:0 , N- 2l8 :2:: 0 it is possible to

write, by the formula

(4) v(x) = [1- H(x)]19N-2ts(x), N- 2l8 :2::0

where 9N-2ts is any polynomial of degree N- 2l8. The fact that (4) gives v = VN E 9Jl:0 is trivial. The converse fact will

be considered here only in the case l = 1, when boundary conditions are reduced to one equation: via = 0.

It affirms that the polynomial v(x) of degree N equals zero there where the polynomial 1 - H(x) is zero. We can mean that the last one is irreducible. Then, by a known fundamental theorem of Algebraic Geometry [4], the polynomial 1 - H divides v N, i.e.

( 4') vN(x) = [1- H(x)]9N-2s(x), N- 28:2::0.

We do not give here the proof for l > 1. Now we can write relation (2") as follows:

(2111) J(LU)9N-2ts(x)[1- h(x)F dx = 0 for all 9N-2t81 N- 2l8 2: 0.

n

We wish to consider the integral f f(x)g(x)[1- H(x)Jl dx =(!,g) as scalar n

product of the functions f, g on D with weight (1 - Hjl. Note that 1 -H(x) > 0, X En.

Let us consider the case 8 = 1. Let P = PN be a given polynomial and N < 2l, then U = PN is the solution of boundary value problem (1): L(PN) = 0, P~")la = P~")la·

Now let N 2: 21. The polynomial U = UN, for which minimum (2) is attained, satisfies (2"'), 8 = 1. Thus the polynomial LU of degree N-:- 21 is orthogonal on D with weight (1 - H) 1 to all polynomials QN_21 of the same degree N - 21. It shows that LU = 0. Therefore it is proved that for

BOUNDARY VALUE PROBLEM FOR ALGEBRAIC POLYNOMIALS 369

any P = PN (N = 0, 1, 2, ... ) the solution of problem (1) is an algebraic polynomial U = UN of degree N. The part 1) of the theorem is proved.

Now we consider the case s > 1. Let AN be the space of polynomials PN of a given degree N. It is possible to prove that

(5) (N 2: 21)

1.e. the operator L takes AN not only to but onto AN -2l·

We represent AN_21 as an orthogonal sum (with weight (1- H) 1):

(6) (N 2: 2/s),

where (for s > 1) N is a nontrivial subspace. Let z = zN_21 E A', z of 0. It follows from (5) the existence of a

polynomial U =UN of degree N, for which LUN = z of 0. Then from (6) it follows that at L(U N) is orthogonal (with weight (1- H) 1) to all polynomials (9)N-E:ff> i.e. U =UN satisfies (2"') where we mean U E 001u. But now from (3) we have:

min E(<p) = E(U) U E 001u <PE!JJlu

and L(U) of 0.

Therefore for s > 1 we discovered an infinite set of polynomials PN, for which the solutions of boundary value problem (1) are not algebraic polynomials of degree N. The part 2) of the theorem is proved, too. D

The results obtained above give reason to assume that it is possible to expand the solutions of the considered boundary value problem in series of algebraical polynomials, which are themselves solutions of the problem. But the details of this question are not the subject of the present note.

REFERENCES

[1] S.L. Sobolev, Partial differential equations of mathematical physics, Pergamon Press, Oxford 1964.

[2] E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971.

[3] S.M. Nikolskii, A variational problem for equations of elliptic type with degeneration on the boundary, Trudy Matem. Inst. AN SSSR, 150 (1979), 212-238 (in Russian).

[4] l.R. Shafarevich, Algebraic geometry, Moscow State University Press, Moscow, 1968 (in Russian).


VOLUME 8

2001, NO 3-4 PP. 371-381

AVERAGING IN DIFFERENTIAL EQUATIONS WITH HUKUHARA DERIVATIVE AND DELAY

V.A. PLOTNIKOV ' AND P.I. RASHKOV t

Abstract. In the paper differential equations with multivalued right-hand sides and solutions are considered with a derivative in the sense of Hukuhara and delay. Theorems are proved for the justification of a general scheme of partial averaging in such differential equations and systems.

The class of differential equations with multivalued right-hand sides and multi valued solutions [1] is a very interesting object for investigations because of its connection with differential inclusions and problems of control.

The notion of a derivative of a multivalued map is introduced for the first time by Hukuhara [2]. Equations of the type

DhX(t) = F(t,X(t))

with Hukuhara derivative Dh are considered by De Blasi and others in [3,4]. They proved theorems of existence and uniqueness, justified the Euler's method and established the equivalence of such an equation with a corresponding integral one.

In his book [5] Tolstonogov discussed in details properties of the bunch of the usual solutions of a differential inclusion and established that it is contained in the multivalued solution of the differential equation with Hukuhara derivative generated by the inclusion.

M. Kisielewicz [6] is the first who considered the problem of averaging in differential equations with Hukuhara derivative and a small parameter.

Different schemes of averaging in differential inclusions with Hukuhara derivative and in more general objects called quasidifferential equations in a metric space are investigated by Plotnikov and Plotnikova [7,8].

* Odessa State University, 2 Petra Velikogo str., 270100 Odessa, Ukraine t A. Kanchev University of Russe, 8 Studentska str., 7017 Russe, Bulgaria

371

372 V.A. PLOTNIKOV AND P.l. RASHKOV

The paper [9] ofT. Janiak and E. Luczak-Kumorek was the first which deals with the averaging method for differential equations with Hukuhara derivative and delay. They justified this method for the case of asymptotically small delay.

In [10] we considered a general type of differential equations with Hukuhara derivative and delays. For them we proved theorems for existence and continuous dependence of the initial functions which are analogous to the corresponding theorems for the equation in the space Rn [11 ,12,13]. In the same paper we proved a theorem for justificating the method of averaging without the assumption that the delay is asymptotically small.

In the present paper we justify a general scheme of partial averaging in differential equations with Hukuhara derivative and delay. The main result is Theorem 4 which is a new one for this type of equations. It follows some ideas of the first author [14] and is based on Theorem 3.

Now we turn to the main exposition. Let comp (Rn) (conv(Rn)) be a space, consisting of all nonempty com

pact (and convex) subsets of Rn having Hausdorff metric H(., .), defined by the formula

H(A, B) = min{r 2 OIA c Br(B), B c Br(A)},

where Sd(C) is a closed d- neighbourhood of a compact set C c Rn. The sets comp (Rn) and conv(Rn) are complete and separable but are

not linear [1].

DEFINITION 1 [2]. Hukuhara difference C = A!!.B of two sets A, BE

conv(Rn) (if it exists) is such a set C E conv(Rn) that A = B + C, where B + c = {b + c I bE B, c E C}.

The basic notion of our work is that of Hukuhara derivative.

DEFINITION 2 [2]. A multivalued map F(.) : R1 -7 conv(R) is called Hukuhara differentiable in the point t0 if there exists a set DhF(t0 ) E

(Rn) h th t th 1· •t 1. F(to+L'>t)l!.F(to) d 1. F(t0 )frF(to-L'>t) conv sue a e tmi s 1m L'>t an 1m L'>t L'>t-*O+ L'>t .... o+

exist and are equal to DhF(to). The set DhF(t0 ) is called Hukuhara derivative ofF in the point t 0 .

We note that in the definition of Hukuhara derivative it is understood that the differences F(t0 )!!.F(to- ~t) and F(t0 + ~t)!!.F(t0) exist for all ~t sufficiently small.

Let us consider differential equations with a multivalued right-hand side F(t,x) and multivalued solutions X(t)

DIFFERENTIAL EQ. WITH HUKUHARA DERIVATIVE AND DELAY 373

(1) DhX(t) = F(t,X(t)), X(t0 ) = X 0,

where F : R1 x conv(Rn) -+ conv(Rn), X 0 E conv(Rn), X : R1 -+ conv(Rn) is absolutely continuous and satisfies (1) almost everywhere.

Theorems of existence and uniqueness of the solution are proved in [3,4],

as well as the equivalence of (1) with the integral equation

t

(2) X(t) = X 0 + j F(s,X(s))ds.

to

In [5] it was established that the bunch of the usual solutions of the differential inclusion

x E <J>(t, x), x(t0 ) = Xo

is contained in the multi valued solution of the differential equation with Hukuhara derivative

DhX(t) = <J>(t,X(t)), X(to) = xo,

which is generated by the differential inclusion. In [6] the method of averaging was justified for the differential equation

DhX(t) = cF(t, X(t)).

In [7] differential inclusions with Hukuhara derivative were considered, in [8] the connection is given between differential equations and inclusions with Hukuhara derivative and quasidifferential equations in metric spaces. In [9] the averaging method is justified for differential equations with Hukuhara derivative and asymptotically small delay.

It is obvious that the differential equations in Rn are obtained as a partial case of differential equations with Hukuhara derivative when the function F(t,x) is singlevalued and X 0 = {x0 }. In connection with this all specific peculiarities of the solutions of differential equations with delay in the space Rn are valid for differential equations with Hukuhara derivative.

However, in differential equations with Hukuhara derivative own peculiarities may appear, connected with the multivaluality. For example, for the solutions X 1 (t) and X2 (t), corresponding to the initial functions <J> 1 (t) and <J> 2 (t) the condition X1 (t) c X 2(t) may hold in the interval [t1 , t 2]. This is not an analogue of the solutions' sticking as X 1 (t) f X2(t).

374 V.A. PLOTNIKOV AND P.L RASHKOV

Let us consider differential equations with delay

(3) DhX(t) = F(t, X(t), X(t- r1(t)), ... , X(t- Tm(t))),

X(t) = (t), tEEt,,

where Eta = uz;1 E~), E~) are sets consisting of the points t0 and those values t r;(t) for which t- r;(t) < t0 when t;:::: t0, : R1 -+ conv(Rn).

For completeness we formulate theorems of existence and uniqueness and of continuous dependence on the initial data from [10].

THEOREM 1. Let F be a continuous function in a neighbourhood of the point (to, ( to), (to- r1(t0)), ... ,( to- Tm(to)), satisfying the Lipschitz condition in all variables starting with the second one, with a constant >.; let the initial function (t) be continuous in E1, and all functions r;(t) be continuous for t0 :S t :S t0 + L (L > 0) and nonnegative. Then there exists a unique solution X(t) of the basic initial problem for the equation (3) for t0 :S t :S t0 + o, where o is sufficiently small.

THEOREM 2. Let all the conditions of Theorem 1 be fullfilled. Then the solution of (3) is continuously dependent in the space Co of the initial functions and from H(1(t), 2(t)) :S 8, 8 > 0, t E E1, it follows that

(4)

Let us consider the following differential equation with Hukuhara deriva-tive:

(5)

X (s,c-) = (s,c), -T :S S :S 0,

where c > 0 is a small parameter, t E [0, Lc-1], L- a constant.

The following partially averaged equation corresponds to ( 5):

(6) DhY (t,c) = cF0 (t, Y (t,c), Y (t- r,c-)),

Y (s,c) = (s,c), -T :S S :S 0,

where

The corresponding scheme of averaging is justified by the following

THEOREM 3. Let in the domain Q = {t 2 0, X, ZED C conv (Rn)} the following conditions hold:

1) the functions F (t, X, Z), F0 (t, X, Z) and iP (s, c:) fulfill the conditions of Theorem 1 and there exists a constant M such that

IF (t, X, Z)l :S M, IF0 (t, X, Z)l::; M, where IAI = H ({0} ,A) ,A E comp(Rn); 2) the limit (7) exists for every X, Z E D; 3) there exists a solution of the system (6) for 0 < c: ::; a and together

with a p-neighbourhood it is in the domain D fortE [0, L*c:- 1], where L* is a constant.

Then for every ry > 0 and 0 < L ::; L * there exists 0 < c:0 ::; a such that for· c: E [0, c: 0] and t E [0, Lc:-1]

(8) H (X (t,c:), Y (t,c:))::; ry.

Proof. We consider the differential equation

(9) DhZ(t,c:)=c:F(t,Z(t,c:),Z(t,c:)),Z(O,c:)=iP(O,c:).

Due to [3, 4] from the differential equations (9) , (5) we come to the corresponding integral equations and get

(10) H(X(t,c:),Z(t,c:))::;

:ScH (/ F(s,X(s,c),X(s-r,c:))ds,j F(s,Z(s,c:),Z(s,c:))ds).

For t E [0, r] taking into account the boundedness of the function F (s, X, Z) we obtain

(11) H (X (t,c:), Z (t,c:))::; c:2Mr.

376 V.A. PLOTNIKOV AND P.I. RASHKOV

For t E ( r, Lc:-1) we have from (10) the following:

(12) H (X (t, c:), Z (t, c:)) ~

~ c:H (/ F(s,X (s,c:) ,X(s- r,c:))ds.J F(s,X (s,c:),X (s,c:))ds) +

(13)

(14)

+c:H (/ F(s,X (s,c:) ,X (s,c:))ds.J F(s,Z(s,c:) ,Z(s,c:))ds) ~

t t

~ c:>..j H(X(s-r,c:),X(s,c:))ds+2c:>..j H(X(s,c:),Z(s,c:))ds. 0 0

Obviously

t J H(X(s-r,c:),X(s,c:))ds~MrL. 0

Thus, using the Gronwall-Beilman's lemma, we get from (12) that

H (X (t, c:), Z (t,c:)) ~ c:MrL exp (c:2>..t).

Analogously for equations (6) and the equation

(15) DhW (t,c:) = c:F0 (t, W (t,c:), W (t,c:)), W (O,c:) = <l? (O,c:)

we have

(16) H (Y (t,c:), W (t,c:)) ~ c:MrLexp (c:2>..t).

Equation (15) is a partial averaging for equation (9). According to [14, p.ll2] for every 1) and L E [0, L*] there exists c: 1 E [0, r]

such that for c: E [0, c: 1] and t E [0, Lc:-1] the following estimate holds:

(17) H (W (t,c:), Z (t,c:)) ~ 1)/2.

From (14), (16), (17) for c:0 = min { c:l, 1)/ (4Mr L exp (2>..L))} we get the estimate (8), which ends the proof. 0

(18)


Now we shall consider another a more general scheme of averaging. Let us consider the differential equation

DhX (t,c) = cF (t,X (t,c), X (t- T1 ,c)), X (t T2/c,c)

X (s,c) = <l? (s,c),

In correspondence to equation (18) we put the following partially averaged equation

(19) Y (s,c) = <l? (s,c),

where

(20) t~~ H ( ~ l F (t, X, Y, Z) dt, ~ l F 0 (t, X, Y, Z) dt) = 0.

The justification of this scheme is the contents of the following THEOREM 4. Let in the domain Q = {t 2: 0, X, Y, Z E conv(Rn)} the following conditions hold:

1) the functions F (t, X, Y, Z), F 0 (t, X, Y, Z) are continuous, uniformly bounded by the constant M and fulfill the Lipschitz condition in all variables starting with the second one with a constant >-;

2) the initial function <l? (s,c) is continuous and

H ( <l? ( s', E) , q; ( s", E)) ::; E), ( s' - 9") , q; (s,t:) ED, -T2jc::; s::; 0;

3) there exists the limit (20) for every X, Y, Z E D; 4) the solution of equation (19) exists forE E [0, T] and together with a

ii-neighbourhood it is in D fort E [0, L*c1], L* is a constant. Then for every 'T/ > 0 and 0 < L ::; L' there exists 0 < t:0 ::; T such that

forE E (0, c0 ) and t E [0, Lc1] the following unequality holds:

(21) H(X(t,c),Y(t,c)):Sry.


Proof. We consider a solution of the systems (18), (19) on the interval [0, 72/c]:

(22) DhX1 (t,c) = cF (t, X 1 (t, c)' X 1 (t- 7), c)' ~ (t- 72/c,c)) '

X1 (0,c)=~(O,c);

Y1 (O,c) = ~(O,c).

According to Theorem 3 for every ry1 > 0 there exists c1 E (0, O"j such that forcE (0, ci} and t E (0, 72/c] the following estimate is valid:

(24) H (X1 (t,c), Y1 (t,c)) :S 1Jl·

Now we consider on [72/c, 272/c] the solutions Y 2 (s, c) and Z 2 (s, c:) of the system (19) with initial conditions

Y 2 (s- 72/2,c:) = Y1 (s,c:),

Z 2 (s- 72/2, c:) = X 1 (s, c:).

Due to Theorem 2 for every ry2 > 0 there exists 6 > 0 such that for H (Y1 (s, c:), X 1 (s, c:)) ::; 6 we have

(25)

Now we consider the solutions X 2 (s,c:) and Z 2 (s,c:) of the systems (18), (19) on the interval h/c:, 272/c:] with initial conditions

According to Theorem 3 for every ry2 > 0 there exists for c: E (0, c:;] and t E h/c:, 272/c:] the following estimate holds

(26)

We choose c~ E (0, O"j such that 7)1 :s; o for E E (0, c:~J. Then for c:2 = min{c:~,c:~J from (25), (26) we get

Let k < -"- :s; k + 1. Then after k steps for c:0 = min1<i<k c; we come to T2 --

the conclusion of the theorem. D

REMARK 1. If the function F 0 (t, X, Y, Z) does not depend explicitly on t, relation (20) means the existence of an average, i.e.

T

F 0 (X, Y, Z) = lim Tl J F 0 (t, X, Y, Z) dt. T-+oo

0

In this case the full averaging scheme justification follows from Theorem 4.

Now we consider the system of differential equations with Hukuhara derivative

(27) D,Xi (t,c:) = c:F; (t, X (t,c:), X (t- 71 , c), X (t- 72/c,c)),

xi (s, c) =if> (s, c) - 72/c :s; s :s; 0, i = 1, 2, ... , m,

where X= (X1o X2, ... , Xm), X E DC W = (conv (Rn') xconv (Rn') x ... xconv (Rnm )) , F = (Ft, Fz, ... , Fm), F (t, X, Y,Z) = F 1 (t, X, Y, Z) + F 2 (t, X, Y, Z), if>= (if>t,if>z, ... ,if>m) ·

We put the following partially averaged system in correspondence to system (27) :

(28) Dhli (t, c)= cF;0 (t, Y (t,c), Y (t- 71,c), Y (t- 72/c,c)),

Y; (s, c) = <l'> (s, c)

where

0 ( 1 -F t,X, Y,Z) = F (t,X, Y,Z) + F 2 (X,Z),

T

(29) F 2 (X, Z) = lim Tl J F 2 (t, X, X, Z) dt. 'T-+oo

0


The following result gives a justification of the above scheme of averaging.

THEOREM 5. Let in the domain Q = Q = { t 2: 0, X, Y, Z E D C W} the following conditions hold:

1) the function F (t, X, Y, Z) is continous, uniformly bounded by a constant M and fulfills the Lipschitz condition with a constant A in all variables starting with the second one;

2) the initial function ill ( s, c) is continous and

H (ill (s', c), ill (s", c)) ~ cA Is'- s"l, ill (s, c) ED, -T2/c ~ s ~ 0;

3) there exists the limit (29) for all X, Z E D; 4) the solution of the system (28) for c E (0, O"j exists and together with

a g-neighbourhood is in the domain D fortE [0, Lc-1].

Then for 'fJ > 0 there exists c0 E (0,0") such that for c E (O,c0] and t E (0, c 1

] the following estimate holds:

H (X (t,c), Y (t,c)) ~ 'f/·

The validity of Theorem 5 follows from Theorem 4 as from the fulfilment of conditions (29) the fulfilment of conditions (20) follows.

REFERENCES

[1] U.G.Borisovicz, R.D.Gelman, A.D.Myshkis, and V.V.Obuchovskii, Multivalued maps, VINITY, Itogi Nauki i Techniki, 19 (1982), 127-231 (in Russian).

[2] M.Hukuhara, Sur !'application mesurables dont Ia valeur est un compact convexe, Func. Ekvacioj, 10 (1967), 43-66.

[3] A.J.Brandao Lopes Pinto, F.S.De Blasi, and F.Iervolino, Uniqueness and existence theorems for differential equations with compact convex valued solutions, Boll. U.M.I., 4 (1970), 534-538.

[4] F.S.De Blasi and F.Iervolino, Equazioni differentiali con soluzioni a valore compatto convesso, Boll. U.M.I., 2, 4-5 (1969), 491-501.

[5] A.A.Tolstonogov, Differential inclusions in banach space, Novosibirsk, Nauka, 1986 (in Russian).

[6] M.Kisielewicz, Method of averaging for differential equations with compact convex valued solutions, Rend. Math., 9, 3 (1976), 397-408.

[7] A.V.Plotnikov, Averaging of differential inclusions with Hukuhara derivative, Ukr. Mat. Zurn., 41, 1 (1989), 121-125 (in Russian).

[8] V .A.Plotnikov and L.I.Plotnikova, Averaging of quasi differential equations in a metric space, Differenzialnie Uravnenija, 31, 10 (1995), 1678-1683 (in Russian).

[9] T.Janiak and E.Luczak-Kumorek, Averaging of neutral differential inclusions with unbounded right-hand side, Discuss. Math., 12 (1992), 67-74 (1993).


[10] V.A.Plotnikov and P.I.Rashkov, Existence, continuous dependence and averaging in differential equations with Hukuhara derivative and delay, Mathematics and Education in Mathematics, Proceedings of 26-th Spring Conf. of UBM, Sofia, (1997) 179-184.

[11] A.D.Myshkis, General theory of differential equations with retarded argument, A.M.S. Translations, series I, 4, American Mathematical Society, Providence, 1962.

[12] ,J.K.Hale and S.M.Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993.

[13] L.E.El'sgol'tz and S.B.Norkin, Intorduction to the theory of differential equations with deviating argument, Academic Press, New York, 1973.

[14] V.A.Plotnikov, A.V.Plotnikov, and A.N.Vitjuk, Differential equations with multivalued right-hand side, asymptotic methods, Astra-Print, Odessa, 1999 (in Russian).


VOLUME 8 2001, NO 3-4 PP. 383-393

A FOURTH ORDER PDE AND ITS APPLICATION IN CONFORMAL GEOMETRY

J. QING '

1. Introduction. In this talk we discuss locally conformally flat manifolds. We first recall that, in the theory of complete surfaces of finite total curvature, Cohn-Vossen [8] showed that if the Gauss curvature of a complete analytic metric is absolutely integrable, then

(1) J KdA :S 21rx,

where x is the Euler number of the surface. Huber [11] extended this inequality to metrics with weaker regularity and proved that such surface can be conformally compactified by adjoining a finite number of points. For such surfaces the deficit in formula (1) has an interpretation as an isoperimetric constant. One may represent each end conformally as R2 \ D for some compact set D and consider the following isoperimetric ratio:

. £2(r) IJ= hm ,

r-+oo 47r A(r) (2)

where L(r) is the length of the boundary circle &Br = {lxl = r}, and A(r) is the area of the annular region B(r) \D. For a fairly large class of complete surfaces called surfaces with normal metrics, Finn [10] showed that,

(3) x(M)- _1:_ j KdvM = L vJ, 27f

M

where the sum is taken over each end of M.

' Department of Mathematics, University of California, Santa Cruz

383

384 J. QING

Let us concentrate on 4-manifolds. We introduce the fourth order curvature invariant Q. For conformal geometry in dimension four, the so-called Paneitz operator

(4) P = A 2 + o G Rg - 2Ric) d,

where o denotes the divergence, dis the differential, R is the scalar curvature, and Ric is the Ricci tensor, plays the same role as the Laplacian in dimension two (cf. [12], [2], [6], for example). The Paneitz operator enjoys the following invariance property under conformal change of metric g = e2wg0 ; the Paneitz operator transforms by P9 = e-4w P90 • The Paneitz operator defines a natural fourth order curvature invariant Q: for the conformal metric g = e2wg0 ,

(5)

where

(6) 1 ( 1 2 2) Q = 6 -!:>.R + 4R - 3IEI

and E is the traceless Ricci tensor. The Q curvature invariant is related to the Chern-Gauss-Bonnet integral in dimension four:

(7) x(M) = 8: 2 J (l~l2

+ Q) dV M

where W is the Weyl tensor and M is a compact, closed 4-manifold. More generally, when the manifold has a boundary, Chang and Qing [3] have defined a boundary operator P3 and its associated boundary curvature invariant T:

(8)

Then the Chern-Gauss-Bonnet integral is supplemented by

(9) x(M) = 8

: 2 J ( 1~1 2

+ Q) dV + 4: 2 J (L + T)dcr, M oM

where Ldcr is a pointwise conformal invariant. In the simplest case when M = R4 we first showed that:

A FOURTH ORDER PDE AND ITS APPLICATION 385

THEOREM 1 [4]. Suppose that e2wjdxj 2 on R4 is a complete metric with its Q-curvature absolutely integrable, and suppose that its scalar curvature is nonnegative at infinity. Then

(10) 1- _1_ J Qe4wdx = lim (vol(~Br(0)))1 > 0. 87f2

R' r-too 4(27r2)avo1(Br(O)) -

The argument is based on our careful study of the so-called bi-harmonic functions and the related fourth order partial differential equations. To take the study to general locally conformally flat 4-manifolds, first, we localize arguments in [4] to an end corresponding to a puncture and obtain

THEOREM 2 [5]. Suppose (R4 \ B, e2wjdxj 2) is a conformal complete metric with nonnegative scalar curvature at infinity. If, in addition,

(11) J iQidV < oo,

then

(12)

This formula then allows us to extend the basic result (10) to allow the domain to have a finite number of punctures.

CoROLLARY 1 [5]. Suppose that (M, g) is a complete 4-manifold with only finite many conformally fiat simple ends. Assume also that the scalar curvature is nonnegative at each end, and the Q curvature is absolutely integrable. Then

(13) 1 I k x(M) -32

7[2 • {IWI 2 + 4Q}dvM =I: vi, M t=l

where in the inverted coordinates centered at each end,

(14) _

1. (vol(8Br))~)

1/i - lffi 1 r-+oo 4(27r2)avol(Br \B)

386 J. QING

Let us give a definition of a conformally flat simple end.

DEFINITION 1. Suppose that (M, g) is a complete noncompact 4-manifold such that

k

(15) M=NU{UE;} i==l

where (N,g) is a compact Riemannian manifold with boundary

k

(16) aN= UaE;,

and each E; is a conformally flat simple end of M that is

(17)

for some function w;, where B is the unit ball in R4• Then we say that ( M, g) is a complete 4-manifold with finite conformally flat simple ends.

We then use the Chern-Gauss-Bonnet formula to derive a compactification criterion in analogy with Huber's two dimensional result.

THEOREM 3 [5]. Let (0 c S 4 ,g = e2wg0 ) be a complete conformal metric such that (a) the scalar curvature is bounded between two positive

constants, and I \7 9RI is bounded with respect to g, (b) the Ricci curvature

of the metric g has a lower bound, (c) the Paneitz curvature is absolutely

integrable, i.e.

Then n = S 4 \ {p1, ... ,pk}·

j IQidvg < oo.

!1

An essential ingredient in the above finiteness result is to view the boundary integral in the Chern-Gauss-Bonnet formula as measuring the growth of volume. The finiteness of the Q integral implies a control on the growth of volume, which can only accommodate the growth of a finite number of ends.

As a consequence we can, for example, classify solutions of the equation Q = constant in the following

COROLLARY 2. lf(D C S 4 ,e2wg0 ) is a complete conformal metric satisfying conditions (a) (b) (c) of Theorem 3 and in addition Q =Constant, then there are only two possibilities:

1. (DC S4, e2wgo) =(54, go);

2. (DC 54, e2wg0) = (R4

\ {0}, I~\\

To deal with non-simply connected 4-manifolds, we recall that in a study of locally conformally flat structures with positive scalar curvature, SchoenYau [16] have proved that the holonomy cover of such manifolds can be conformally embedded into spheres with boundaries of small Hausdorff dimension. Thus our compactification criterion applies to simply connected manifolds, for which conditions (a), (b), and (c) hold. In fact, since the argument localizes to each end, we can dispense with simple connectivity assumption provided we are willing to assume the geometric finiteness for the honolomy representation of the fundamental group of the 4-manifold. For convenience, we will follow the terminology of [14].

Suppose that (M, g) is a locally conformally flat manifold with positive s:alar curvature. Then, by the result of Schoen-Yau [16], the universal cover M can be embedded as a domain into the 4-sphere. Hence the fundamental group r acts on 5 4 as a discrete group of conformal transformations with a. domain of discontinuity D(f), which contains M. The limit set L consists of accumulation points of orbits of r. The discrete group r also acts as hyperbolic isometrics on the interior B of 54 We recall the following definitions of limit points. A point p E 5 4 is called a. conical limit point of the group f if there is a point x E B, a sequence {gi} C f, a hyperbolic ray 1 in B ending at p, and a. positive number r such that {gi(x)} converges top within hyperbolic distance r from the geodesic 1. A point p is a cusped limit point of a discrete group r if it is a fixed point of a parabolic element of r that has a cusped region. To explain the notion of a cusped region, we identify 54

with R4 , conjugate the point p to infinity in the upper half space R~ in R5 ,

and consider the stabilizer r 00 of oo. r 00 is a. discrete subgroup of isometrics of R4 of rank 0 < m ~ 4. Let E be the maximal r 00-invariant subspace such that E/f oo is compact. Denote by N a neighbourhood of E in R~, and set U = R~ \ N. Then U is an open roo invariant subset of R~. The set U is said to be a cusped region for r based at oo if and only if for all g in r \ r 00 ,

we have U n gU = 0.

DEFINITION 2. A convex polyhedron Pin the hyperbolic space is called geometrically finite if and only if for each point x E P n 54 there is an open Euclidean neighbourhood of x that meets only the faces of P incident with x. A discrete group of conformal transformations of 54 is called geometrically finite if and only if r has a geometrically finite convex fundamental polyhedron.

388 J. QING

An important aspect of geometrically finite Kleinian groups is that its limit set consists only of conical limit points and cusped limit points ([14], Theorem 12.3.5, p. 512.) Now we are ready to state and prove our main theorem in this talk.

THEOREM 4 [5]. Suppose that M is a locally conformally fiat and complete 4-manifold, which satisfies conditions (a), {b), (c) in the previous Theorem 3. And, in addition, we assume that (d) the holonomy representation of the fundamental group r is a geometrically finite Kleinian group without torsion.

Then M = M \ {p1 , ... ,pk}, where M is a compact conformally fiat 4-manifold.

2. Proof of Theorem 3. In the following we would like to give an outline of the proof of Theorem 3. We remark that given any simply connected, locally conformally flat, complete manifold M of dimension n 2: 3, there always exists an immersion iP : M -+ sn such that the locally conformally flat structure of M is induced by iP. This immersion iP is called the developing map of M. By a well-known result of Schoen and Yau cf. [16, Chapter 6], under the further assumption that the scalar curvature R9 2: 0, iP is injective. Therefore any such manifold can be considered as a subdomain of sn with a

4 complete metric g = vn-2 gc, where gc is the standard metric on the sphere sn. Combining this result with Theorem 3, we obtain:

COROLLARY 3. Suppose that M is a simply connected, locally conformally fiat, and complete 4-manifold satisfying conditions (a), {b), and (c) as in Theorem 3. Then M is conformally equivalent to S4 \ {PiH~1 for finite many points {Pi}f~1 on S 4

•

Suppose that M is a subdomain of sn, for convenience, we choose a point P in M and use a stereographic projection, which maps sn to Rn and P to

4 infinity; then we may identify M with M = (!1, u= ldxl 2

), where !1 C Rn and jdxj2 is the standard metric of Rn. In the following we will estimate the size of the conformal factor u(x) for x E !1 in terms of the Euclidean distance d(x) = distance(x, 8!1). First we have the following lower bound from [16, Theorem 2.12, Chapter VI]:

4 LEMMA 1. Let M = (!1, g = un-,jdxj 2 ), where !1 C Rn; and suppose

that (a) IRI and I \7 9 RI both bounded with respect to g, (b) the Ricci cur-vature has a lower bound.

Then ther-e exists a constant C > 0 such that

(18) n-2

u(x) 2: Cd(x)--2

for all X E Q.

We remark that in the statement of Theorem 2.12, Chapter VI in [16], a stronger assumption that M has bounded curvature is listed for the above result. But it is clear from the proof (e.g. applying method of gradient estimate), that assumptions as (a) and (b) are sufficient for the conclusion.

It turns out that in the case when M = (D,u2 ldxl2), where D c R4 and

satisfies assumptions (a) and (c) in Theorem 3, we can also establish the upper bound of u in terms of the distance function d.

LEMMA 2. Suppose that M = (D, u2 ldxl2) is a complete manifold such that

(a) its scalar curvature R satisfies 0 < R0 ::; R::; R 1 and l\7 Rl 9 ::; C, wher-e Ro, R1, C ar-e constants, and

(c) J 1Qin4dx < oo. Then there exists some constant C such that ()

(19) u(x) ::; Cd(xt 1

for all X E D.

Our proof of the above lemma uses a blow-up argument, which in the case when the scalar curvature function R is a constant has been applied by Schoen (presented in [13]) to obtain the same upper bound (19). Thus what we have done here is replacing the constant scalar condition by the integral bound of the Paneitz curvature and condition (a).

The proof of Lemma 2 depends on the following simple result, which is a special case of Theorem 4.1 in [4].

LEMMA 3. On (R4,u2ldxl 2), the only metric with Q = 0 and R 2:0 at infinity is isometric to (R4, ldxl2 ).

We will now estimate the size of the integral of Q over a subset of D in terms of the integral of the boundary curvature T (as defined in the introduction) via the Chern-Gauss-Bonnet formula. To do so, we will first derive a formula for the boundary integral. We will use the following notations. We consider M = (D, u2 ldxl2 ), where D c R\ and denote the level set for the conformal factor u = ew by

(20) U>. = {x: 1::; u::; A}, and S>. = {x: n =A}.

Also, On denotes the normal derivative (chosen so that ~~ 2: 0).

390 J. QING

LEMMA 4. Suppose that M = (n, u2g0 ) is a complete Riemannian manifold. Then on the level set S>., where ,\ is a regular value for u, we have:

(21)

Finally, we state a simple covering lemma, which we will use in the proof of Theorem 3.

LEMMA 5. Suppose that A is a compact subset of R4 • Then

(22) l{x: dist(x,A) = s}l2 Ns3,

for any N > 0 if H 0(A) = oo, where Hf! denotes the fJ dimensional Hausdorff measure of the set.

We now sketch the proof of Theorem 3. We identify (M, g) with (!2, u2 ldxl2 ) for some subset n of R4 as before. Notice that all ends of Mare in a bounded region inside n. Applying integration by parts, we get

(23) J Qe4wdx = J 6.2wdx = - J 8n6.wdrJ + J 8n6.WdrJ U;.. U>.. 81 8;..,

Applying formula (19) in Lemma 2, we obtain

-J Qe4wdx- J 8n6.wdrJ =-J 8n6.wdrJ =

u, s, s,

=Add;..( J (8nw) 3drJ + J J(8nw)e2wdrJ + 2 J JIV'ul 2dx )+ ~ ~ ~

(24)


where V (,\) is defined as:

(25) V(>.) = J (8nw) 3 dcr + J J(8nw)e2wdcr + 2 J J!'Vu! 2dx. s, s, u,

We recall the scalar equation

(26)

inn. Thus,

J J!'Vu! 2dx 2: ]0 J !'Vu! 2

dx 2: u>. u>-

(27)

where J 2: J0 > 0 as assumed in (a). And

(28) V(,\) 2: 215 j u4dx. u,

To estimate the growth of V, we use the lower and upper bound estimates of the conformal factor u as in Lemma 1 and Lemma 2. Thus we may replace the region U;. by

(29)

Therefore we have

(30)

by the co-area formula. Hence,

(31)

c,

V(>.) 2: C.! !{x: d(x, on) = s }!s- 4ds.

'22. A

392 J. QING

We now estimate the size of the set an by Lemma 5. If H 0 measure is infinite, then

(32) l{x: d(x,an) = s}l2: Ns3

for any number N > 0. Hence,

(33)

c,

V(-X) 2: N J ~ds = Nlog-X- C.

£2. '

Thus we conclude that there exists at least a sequence of A; -+ oo as i -+ oo such that Ai are all regular values (due to Sard's theorem) and

(34)

for any number N > 0. But in view of equality (24), this contradicts our assumption (c) that Q is integrable. We have thus finished the proof of the theorem.

REFERENCES

[1] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, GTM 137, Springer-Verlag, 1992.

[2] T. Branson, S-Y. A. Chang and P. C. Yang, Estimates and extremal problems for the log-determinant on 4-manifolds, Communication Math. Physics, 149, 2 (1992), 241-262.

[3] S.-Y. A. Chang and J. Qing, The zeta functional determinants on four manifolds with boundary I- the formula, J. Funct. Anal., 147, 2 (1997), 327-362.

[4] S.-Y. A. Chang, P. C. Yang and J. Qing, On the Chern-Gauss-Bonnet integral for conformal metrics on R<, to appear in Duke J. of Math.

[5] S.-Y. A. Chang, P. C. Yang and J. Qing, Compactification on a class of conformally fiat 4-manifolds, Preprint, 1999.

[6] S-Y.A. Chang and P. C. Yang, Extremal metrics of zeta function determinants on 4-manifolds, Annals of Math., 142 (1995), 171-212.

[7] J. Cheeger and M. Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, Differential geometry and complex analysis, Springer, Berlin-New York (1985), 115-154.

[8] S. Cohn-Vossen, Kiirzeste Wege und Totalkriimmung auf Fliichen, Compositio Math., 2 (1935), 69-133.

[9] K. J. Falconer, The geometry of fractal sets+, Cambridge University Press, 1985. [10] R. Finn, On a class of conformal metrics, with application to differential geometry

in the large, Comm. Math. Helv., 40 (1965), 1-30. [11] A. Huber, On subharmonic functions and differential geometry in the large, Comm.

Math. Helv., 32 (1957), 13-72.

:-·-~·· '.,


[12] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint, 1983.

[13] D. Pollack, Compactness results for complete metrics of constant positive scalar curvature on subdomains of sn, Indiana Math. J., 42 (1993), 1441-1456.

[14] J. Ratcliff, Foundations of hyperbolic manifolds, Springer Verlag, 1994. [15] R. Schoen, The existence of weak solutions with prescribed singular behaviour for a

conformally scalar equation, Comm. Pure and Appl.Math., 41 (1988), 317-392. [16] R.Schoen and S.T. Yau, Lectures on differential geometry, International Press, 1994.

VOLUME 8

2001, NO 3-4

PP. 395--406

ON THE BOUNDARY VALUE PROBLEM FOR THE ELLIPTIC FUNCTIONAL-DIFFERENTIAL EQUATION WITH

CONTRACTIONS '

L.E. ROSSOVSKU t

Introduction. The elliptic functional differential equations containing transformations of arguments were considered by many authors [2, 4, 5 and others]. The influence of arguments' shifts in the highest derivatives (when these shifts map points of the boundary into the domain) on solvability and regularity of solutions of boundary value problems was studied first by A.L. Skubachevskii. For the introduction to the theory of elliptic differentialdifference equations in bounded domains, see [8].

Boundary value problems for elliptic functional differential equations with contractions were considered in [6,7]. The questions of solvability and regularity of solutions were studied in Sobolev spaces. In particular, it was shown that the problem

(0.1) (x E Q),

(0.2) u(x) = 0 (x E aQ)

with q > 1, Q C qQ C Rn, has a unique generalized solution u E W21(Q)

l for every f E L2 (Q) iff the complex polinomial a(.\) = :Z::: ak.>-k does not

k~o

vanish in the disk 1->-1 < qnf2- 1 Moreover, in this case the solution u E

• The work was partially supported by Russian Foundation of Basic Researches (grant N 99-01-00028) and INTAS (grant N 97-30551).

t Moscow State Aviation Institute, 125871 Russia, Moscow, Volokolamskoe shosse, 4

395

. ,

396 L.E. ROSSOVSKII

wr2 (Q) whenever f E Wf(Q), p = 0, 1, .... Otherwise, there is an infinitedimensional kernel of nonsmooth functions u ¢:. Wi(Q) in (0.1), (0.2).

In the present paper we consider problem (0.1), (0.2) in weighted spaces H~,.( Q), regulating power singularities of solutions at the origin. We show that (0.1), (0.2) is always Fredholm in H~,,(Q) with suitable exponents k and s.

1. Functional Operators. Given q > 1, we assume that Q c JRn is a smooth bounded domain such that

(1.1) QcqQ.

Denote by H~,,(lRn) (k = 0, 1, ... ; s E JR) the supplement of the space C0 (JRn \ {0}) with respect to the norm

and by H~,.( Q) the space of restrictions of functions from H~,s (JRn) to Q. The space H~,,(O) (fJ = Q; JRn) is a Hilbert space with respect to the inner product

(u,v)Hf,,(n) = L lalsk

J lxl2(s-k+Jal) D"'u(x)D"'v(x) dx.

!1

Consider the bounded operator R : Hg,s (JRn) -7 HL (JRn) defined by the formula

Ru(x) = u(q-1x) = u(~1 , ... , Xqn) .

Obviously, the inverse and adjoint operators have the form

(R-1u)(x) = u(qx), (R*u)(x) = qn+2'u(qx),

and therefore the operator R is normal and its spectrum O'(R) C {>- E C: 1>-1 = qnf2+s}. It can easily be checked that O'(R) ={A E C: 1>-1 = qnf2+s}.

Now let us introduce the functional operators A(x, R) with variable coefficients. Let B be a Banach space. The space of all B-valued analytic functions in an open subset fJ c Cis denoted by H(fJ,B). When fJ = {.\ E C : l.\1 < r }, B = Ck(Q) (k = 0, 1, 2, ... ), we write H(fJ, B) = 1i(r, k).

ELLIPTIC FDE WITH CONTRACTIONS 397

Similarly to scalar case, it can be shown that functions a E 1-l(r, k) are sums 00

of power series I; arn(x)>.m with coefficients am E Ck(Q) such that m=O

(1.2) (r' < r).

Besides, am(x) = ,;;,a~m)(x,O); as Mk(r') we can take maxJia(-,>.)ib("Q)· 1>-l=r'

LEMMA 1.1. If a E 1-l(r, k), then the formula

00 1 A(x,R)u(x) = L -1 a~m)(x,D)Rmu(x)

m=Om.

defines a bounded operator A(x, R) : HL(Q) -+ H~ ,(Q) for all k = 0, 1, ... ' '

and s E JR: such that s - k < logq r - n/2.

REMARK. We say that a(x, >.) is a symbol of the operator A(x, R).

Proof. First prove that A(x, R) is bounded in HJ,,(Q). If a is a polynomial of>., then by (1.1) the operator A(x, R) is well defined. For an arbitrary

(X)

a E 1-l(r,k), show that the series I; amRm converges in B(HS,(Q)). Here m=O

am(x) = ,;;,a~m)(x,O). First note that the norm of Rm in B(HS,,(Q)) does not exceed its norm in B(HJ,,(JR:n)), i.e. qm(nf2+s)_ Further, llguiiHg_,(Q) ::;

ll9llc(Q) · llulluf.,(Q)• when g E C(Q), u E HL(Q). Therefore from (1.2) it

I ( nfZ+•)m /2+ follows I arnRmiiB(Hf,,(Q)) :::: Mo(r') rcr for some r' > qn 8

• But then we have

r' IIRIIB(L,(O,T)) :C: Mo(r') , n/2+s ·

r - q

Considering the operator A(x, R) in H~ ,(Q), let us show that A(x, R)u E H~,,(Q) whenever u E H~,,(Q). Then by,the closed graph theorem the operator A(x, R) is bounded in H~,(Q). Using the Leibnitz formula, we have

D" A(x, R)u = L ( ~ ) Acx-iJ(X, q-liJIR)D!iu iJ<oc

where by Aa-iJ(x, q-II'IIR) we denote the operator with the symbol D~-!ia(x, q-1!31,>.). Note that D~-!ia(x, q-1!31,>.) E 1-l(ql!ilr, k -lad+ 1!31). Then, as we have just seen, the operator Aa-iJ(x, q-liJIR) : Hg,s-k+liJI(Q) -+

398 L.E. ROSSOVSKII

Hg,s-k+lf3I(Q) is bounded. Since u E H~,.(Q), we have Df3u E Hg,s-k+lf3I(Q) (as it follows from the definition of the space HL( Q)). Hence

'

Aa-{3(x, q-!3 R)D13u E Hg,s-k+f3(Q) C Hg,s-k+lai(Q)

and therefore D"A(x,R)u E H8,s-k+a(Q), i.e. A(x,R)u E H~,,(Q). This completes the proof. D

Operators with symbols from 1-l(r, k) form algebra A,.,k, which is not commutative. Multiplying corresponding series, we obtain the composi-

oo 00

tion formula. Suppose A(x, R) = I; am(x)Rm, B(x, R) = I; bm(x)Rm, m=O m=O

where a,b E 1-l(r,k). Then the composition AB is the operator C(x,R) = 00

I; Cm(x)Rm with m=O

(1.3) m

em(x) = L a1(x)bm-Aq-1x) j=O

00

(m=O,l, ... ).

It can easily be checked that c(x, >.) = I; cm(x)>.m E 1-l(r, k). m=O

Now let us prove the following invertibility theorem in Ar,k· I

THEOREM 1.1. Let a(x, >.) be the polynomial I; am(x)>.m, am E Ck(Q) m;::::;:;O

(m = 0, ... , l) and 1. a(x, 0) = ao(x) f 0 for all x E Q, 2. a(O, >.) f 0 for all ,\ E <C such that 1>-1 < r. Then there exists the inverse operator A-1(x, R) E A,.,k·

Proof. Without loss of generality it can be assumed that a0 (x) = 1. Let us use composition formula (1.3) for constructing the inverse operator

00

A-1(x,R) = B(x,R) = I; bm(x)Rm. The equality AB =I leads to the m=O

system

(1.4) {

bo(x) = 1,

bm(x) =-%;; a1(x)bm-j(q-1x), m = 1, 2, ....

The equality BA =I gives

(1.4') {

bo(x) = 1,

bm(x) = ~1 aj(q1-mx)bm-j(x), m = 1, 2, ....


Show that both systems define the same sequence of coefficients bm(x), m = 0, 1, .... Introduce them x m matrices Am(x) (x E Q; m = 1, 2, ... ):

a1(x) az(x) am-!(x) am(x) 1 al(q- 1x) am-z(q-1x) am-J(q- 1x)

Am(x·) = 0 1 am-s(q-2x) am-z(q-2x)

0 0 1 a1 ( q-(m+l) X)

We claim that the functions bm(x) in (1.4) and (1.4') can be computed according to the formula

(1.5) bm(x) = (-l)mdetAm(x) (m = 1, 2, ... ).

The proof is by induction over m = 1, 2, .... Form= 1, there is nothing to prove. Assume (1.5) is valid for all natural numbers up to m. Uncovering det Am+! (x) over the first column and using the induction hypothesis, we get

m+l

-am+l(x) =- 2=>1(x)bm+l-J(q-ix), j=l

i.e. (1.4). If we uncover det Am+l(x) over the last row, we have

400 L.E. ROSSOVSKII

m+l

... - am(q-1x)(-1)detA1(x)- am+1(x) =- L aj(q-m-l+jx)bm+!-j(x), j=1

i.e. (1.4'). Clearly, bm E Ck(Q), m = 0, 1, .... The theorem is proved if we show

that for any r' < r there exists Mk = Mk(r') such that

(1.6) (m=1,2, ... ).

Introduce the square l x l matrix

(

( 1-l ) ( 2-l ) G(x) = -a1 q x -a2 ~ x

and put Gm(x) = G(q-mx), Bm(x) = (bm+l(x), ... ,bm+1(x)f, m = 1, 2, .... Then (1.4') is equivalent to Bm = GmBm-1 = GmGm-1 · · · G1Bo. (m = 1, 2, ... ). To prove (1.6), we need the following estimation (1.7) m~ IID"(Gm · · · G!)(x)ll::; c.,(r')r'-m (r' < r; Ia!::; k; m = 1, 2, ... ) xEQ

(in (1. 7) II · II is a matrix norm). Let us prove (1.7) by induction over Ia! = 0, 1, ... , k. It can easily

be checked that m~IIGm(x)- G(O)II ::; cq-m. Applying Lemma 1 from xEQ

Appendix in the situation B = Ctxl(Q), g = G(O), 9m = Gm(x) (by ClXl(Q) we denote the Banach algebra of matrix-valued continuous functions), we get

(r > p( G(O)) ).

Let us estimate the spectral radius p( G(O)) of G(O) in B. The eigenvalues z1, ... , z1 of G ( 0) are the roots of the equation

(1.8)

After substitution z = A -I (1.8) looks as a(O, .\) = 0. By assumptions of the theorem all the roots Zj of (1.8) satisfy the inequality !zj! ::; r-1. Thus, p(G(O)) ::; r-1. Take any positive r' < rand put f = r'-1. Then f > r-1 :::>: p(G(O)), and

m~ IIGm(x) · · · G1 (x)ll ::; c0 (r')r'-m xEQ

(m = 1, 2, ... ).

For lal = 0, estimation (1.7) is proved. Assume (1.7) is valid for all a such that Ia I :S N < k. Consider the derivative 8~, D"( Gm · · · GI) (Ia I = N; t = 1, ... , n). By the Leibnitz formula,

~ D"' (G G ) D"'-"'' oG 1 D"'-"' (G G ) X L.,; m · · · j+! · ~ · j-1 · · • 1 · j=1 u~

Obviously, there exists M > 0 such that IID1 GJ(x)ll :S M (x E Q, hi :S k; j = 1, 2, ... ). Then, using the induction hypothesis, we have

m "'c (r')rfi-mc (r')r' 1-i < L.-J az a-cq _

:S eM [ L Ca2 (r')cc.-a, (r')] mr'l-m :S Cc.•(r')r'-m, az ~0:1 :Sa

where la'l = N + 1. Thus (1.7) is valid for all a such that lal = N + 1. The theorem is proved. D

From Theorem 1.1 and Lemma 1.1 it follows I

THEOREM 1.2. Let a(x, .\.) be the polynomial :Z:: am(x).\.m, am E Ck(Q) m::::::O

(m = 0, ... ,l) and a0 (x) # 0 for all x E Q. I

Then the bounded operator A(x, R) = 2:: am (:r )Rm : H~,s ( Q) -+ H~,( Q) m::;;;;Q

has a bounded inverse if a(O, .\.) # 0 for all .\. E IC such that l.\.1 ::; qnf2+s-k.

402 L.E. ROSSOVSKII

2. The Boundary Value Problem. Consider the boundary value problem

(2.1) .6.(A(x, R)u)(x) = f(x) (x E Q),

(2.2) ulaQ = 0,

l where A(x, R)u(x) = I: am(x)u(q-mx), am E Ck+2 (Q) (m = 0, ... , l), f E

m=O

H;,.(Q) are complex-valued functions. Let us introduce a bounded operator .CR: Htt2 (Q)---+ H;,.(Q) with do

main V(.CR) - {u E H~,~2(Q): ulaQ = 0} by the formula .CRu = .6-(A(x, R)u). We say that u is a solution to problem (2.1), (2.2) if u E V(.CR) and .CRu =f.

Since u E H~,~2(Q), am E Ck+2(Q) (m = 0, ... , l), we have

.6.(A(x, R)u) = A(x, q-2 R).6.u + 2[V' A(x, q-1 R)]V'u + [.6-A(x, R)]u,

where V' or .6. near A mean the corresponding differentiation of the coefficients w.r.t. x. Clearly, Ax;x;(x,q-1R), Ax;(x,R) are bounded in H;,.(Q).

In this section we assume that 1. a0 (x) # 0 for all x E Q;

l 2. a(O, .\) = I: am(0).\1 # 0 for all.\ E IC such that l.\1 ::; qnf2+s-k-2.

m=O

Then a(O, q-2 .\) # 0 (l.\1 ::; qnf2+s-k), and by Theorem 1.2 the operator A(x, q-2 R) in H;,.( Q) has a bounded inverse B(x, R). Applying B(x, R) to the both sides of equation (2.1), we get

.6-u+Ku=g,

where K: H~;t1 (Q)---+ H;,.(Q) is a linear bounded operator, g = B(x, R)f E

H;,8 (Q). Thus, under the above assumptions, the study of boundary value problem (2.1), (2.2) can be reduced to the study of the model equation in ]Rn;

(2.3) Llv = h

v E Hk+2 (1Rn) hE Hk (JRn). 2,s ' 2,s Following [3], we pass in JRn to the spherical coordinates (p, w),

0 < p < oo, w E sn-1, putting

(

n ) 1/2

P = lxl = _L x] , J;!

w; = w;(y!, ... , Yn) (i = 1, ... , n- 1),

ELLIPTIC FDE WITH CONTRACTIONS

X· Y . - _}_ J-

p (j=1, ... ,n),

403

where wi are infinitely differentiable functions (sn-1 is the unit sphere in ocn ). Equation (2.3) in these coordinates will be written in the form

fPv 1 n-1 &2v 1 n-

1 &2v " 2 +- L>i(w) 8 8 . + 2 L aiJ(w) 8 8 + up p ._

1 p w1 p .. _

1 w, w1 J- Z,J-

1 &v 1 n-1 fJv

+-b(w)- +- v b (w)- = h, p &p p2 LJ J ow

J=1 J

where a1(w), ai1(w), b(w), b1(w) E C00 (8n-1). After change of variables

7 = -lnp we get

&Zv n-1 &Zv n-1 &2v &v n-1 &v " 2 +"'ai(w)

0 0 +L::ai1(w)

0 0 +(b(w)+1),-+Lb1(w);:;-=

U7 LJ 7 W· W· W· U7 UW· j=l J i,j=l z J j=l J

((7,w) E Jf!. X sn-1).

As a result of smooth nondegenerate transformations, this equation is elliptic. After Fourier transform over 7 we come to the equation

n-1 ()2- n-1 fJv L::aii(w) 0w;w -i.\Lai(w) 0w - (.A2 +i.A(b(w)+1))v+ i,j=l t J j=l J

(2.4) n-1 <:>-V vV -

+ Lib1(w)0w =F(.A,w),

i=1 J

which is elliptic in sn-l for every .A E <C. We see that hE H~,.,(lf!.n) implies

FE W~(sn- 1 ), and it can be proved the estimate

k +oo+i11

c1\\hiiii},,(!Rn) :S: L J \.AI 2PIIFII~;-r(sn-1) d.\ :S: c211hll~~.,(lll.")' p=O -oo+in

where 71 = -n/2- s + k + 2; c1 , c2 > 0 do not depend on h. Equation (2.4) satisfies the assumptions of [1] in some angle I arg .AI ::; o, l1r - arg .AI ::; o. Therefore we have the a priori estimate

I.AIZ(k+2)llvlll,(s"-1) + llvll~;+'(s"-1) :S: C3 (1>-l 2kiiFIIL(sn-1) + IIFII~,f(s"-1))

404 L.E. ROSSOVSKII

(c3 > 0 is independent on F and A) and existence of a unique solution to (2.4) provided A is large enough.

Let l = C(w, iA, a jaw): w;+2(sn- 1)-+ W~(sn-1 ) denote the bounded operator in the right-hand side of (2.4). Then the inverse operator Z- 1(A) is a finite-meromorphic operator-valued function of A. If there are no poles of Z-1(A) on the line Im A = -n/2- s + k + 2, then by theorem 1, [3] for every function h E HL(JRn) equation (2.3) has a unique solution v E H~t2 (JR.n ),

' ' depending continuously on h. Under the above condition on Z-1 one can prove Fredholm property

of C R constructing right and left regularizers. Let us show, for instance, how to obtain the right regularizer E0 : HL(Q) -+ 'D(CR)· Denote by E 1 : H~ 5 (lR.n) -+ H~t2 (JR.n) the bounded op~rator, giving the solution of

' ' (2.3), by E2 : W~(Q) -+ w;+2(Q) n Wi(Q) the bounded operator, solving the Dirichlet problem for the Poisson equation in Q. Let </Jj E C0 (JR.n), j = 1,2 be a partition of unity in Q such that </J1 is finite in Q and </J1(x) = 1 in some neighbourhood of the origin. Take also a function 'lj;1 E C0 ( Q) such that '!f;1(x) = 1 in supp</J1, and a function 'lj;2 E C0 (JR.n) such that '!f;2 (x) = 1 in supp</J2 and 'lj;2(x)- 0 in a neighbourhood of the origin.

Put E 0 = L; <PjEj'lj;jB(x,R). Then Eo acts from H~ 5 (Q) to 'D(CR), ~1~ '

and

CREof = A(x, q-2 R)(b, + K) L <PjEj'lj;jB(x, R)f = j=1,2

= A(x, q-2 R)b, L <PjEj'lj;jg + Kd = j=l,2

= A(x,q-2 R)g+ Kd = j +K2f.

Compactness of K 1 and K2 in Ht(Q) is a consequence of the compact embedding H~,-;2 (Q) c H~,-; 1 (Q). Slmilarly the left regularizer can be constructed.

Thus we have obtained the following solvability theorem.

THEOREM 2.1. Suppose the following conditions to be fulfilled: 1. ao(x) # 0 for all x E Q;

ELLIPTIC FDE WITH CONTRACTIONS

2. a(O, >.) of 0 for all ,\. E C such that 1>.1 :S qn12+s-k-2 ;

3 . .C-1 (>.) does not have poles on the line Im>. = -n/2- s + k + 2. Then the operator .CR: H;,-;2 (Q) _, H~,,(Q) is Fredholm.

405

The third condition of Theorem 2.1 in two-dimensional case can easily be given in explicit form. The operator C(>.) : w~+2 (S1 ) --+ W~(S1 ) is the ordinary differential operator C(>.) = ,\.2

- £, with the periodic condition

v(>.,w + 21r) = v(>.,w). We see that .C-1 (>.) has poles at the points >.P = ip, p E z. So the third condition in the case n = 2 means that s ~ z.

Appendix. Let B be a Banach algebra, g E B, and p(g) = lim llgnll ~ n-+oo

be the spectral radius of g (II · II = II · liB)· It is well known that spectral radius is continuous from above: if ll.9m gil-+ 0, then lim p(gn) :S p(g).

n-+oo

LEMMA 1. If llgn- gil :S chn (0 :S h < 1), then

Proof. Denote gn- g = tlgn; gngn-1 · · · g1 - gn = Lln· We shall estimate lltlnll· Let us group the remainder Lln = (g+tlgn)(g+Llgn-1) · · · (g+tlg1)

gn into n sums Lln,m (m = 1, ... , n) such that Lln,m contains exactly m cofactors tlgi in each summand.

n i1-l

Lln,m = I: gn-i, (LlgiJ I:

im-1-l

X L gim_,-1-im(Llg;m)gim-1

im=l

Since llgkll :S M(p)tl+l for an arbitrary p > p(g)

we get

Thus,

n i1-l

lltln,mll :S [M(p)jm+l Pn+lcm L L i1=mi2=m-1

(M(p) =max llg- >-ell), 1-"l=i>

im-1-l L hi1 +iz+ ... +im.

im=l

406 L.E. ROSSOVSKII

The last sum allows immediate estimation

n i1-l im-1-l :L :L ... :L hi,+i,+ ... +i,. s i1==mi2=m-l im=l

00 00

< :L hi' :L i1:::;m i2::;:;m-l

But the sequence [ cMg)hh7] m is majorized by a decreasing geometric

progression, so that IILlnJI S M 1(p)j]'. Thus we obtain

From here it follows lim llYn ... 9111 1/n S p. But p > p(g) is arbitrary, and n->oo

the lemma is proved. D

REFERENCES

[1] M.S.Agranovich and M.I.Vishik, Elliptic problems with parameter and general parabolic problems, Uspekhi Mat. Nayk, 19 (1964), 53-161 (in Russian).

[2] A.B.Antonevich, Linear functional equations. Operator approach, Birkhiiuser, Basel-Boston-Berlin, 1995.

[3] V.A.Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obshch. 16 (1967), 209-292 (in Russian).

[4] D.Przeworska-Rolewicz, Equations with Transformed Argument, PWN, Warszawa, 1973.

[5] V.S.Rabinovich, On the solvability of differential-difference equations on ~n and in a half-space, Soviet Axad.Sci. Dokl. Math., 243 (1978), 1498-1502 (in Russian).

[6] L.E.Rossovskii, Coerciveness of functional differential equations, Mat. Zametki, 59 (1996), 103-113 (in Russian).

[7] L.E.Rossovskii and A.L.Skubachevskii, Solvability and regularity of solutions for some .classes of elliptic functional differential equations, The sciences and technique summary VINITI, 66 (to be published in 2000).

[8] A.L.Skubachevskii, Elliptic functional differential equations and applications, Birkhiiuser, Basel-Boston-Berlin, 1997.


VOLUME 8 2001, NO 3-4 PP. 407-424

THE MIXED BOUNDARY VALUE PROBLEM FOR PARABOLIC DIFFERENTIAL-DIFFERENCE EQUATION *

A.L. SKUBACHEVSKU AND R.V. SHAMIN t

Abstract. For the mixed boundary value problem for parabolic differential-difference equation, we prove uniqueness and existence of strong solutions and stability of solutions. The proofs are based on the semigroups theory and on the properties of difference operators in Sobolev spaces.

1. Introduction. We consider the first mixed boundary value problem for parabolic differential-difference equation. Under assumptions of minimal smoothness for initial functions, we prove uniqueness and existence of strong solutions. For sufficiently smooth initial functions, the first mixed problem for differential-difference equation was studied in [1]. We note that parabolic functional differential equations have important aplications to nonlinear optics [2]-[4]. On the other hand, parabolic differential-difference equations are closely connected with nonlocal elliptic and parabolic problems arising in plasma theory [5], [6].

2. Properties of Difference Operators. 1. Everywhere henceforth we shall assume that the following condition is satisfied:

CONDITION 2.1. If n = 1, let Q = (0, d). In the case n ;::: 2, let

Q C JR.n be a bounded domain with a boundary 8Q = U Mi ( i = 1, ... , N0). i

Here Mi are the ( n- 1) -dimensional manifolds of class coo, which are closed and connected in the topology of 8Q. Let in the neighbourhood of each point g E K = 8Q\ U M; the domain Q be diffeomorphic to an n-dimensional

i dihedral angle if n ;::: 3, and to a plane angle if n = 2.

* This paper was carried out with the financial support of INTAS (grant N 97-30551), RFBR (grant N 99-01-00028), and Deutsche Forschunsgemeinschaft

t Moscow State Aviation Institute, 125993 Russia, Moskow, Volokolamskoe shosse 4

407

408 A.L. SKUBACHEVSKII AND R.V. SHAMIN

We introduce a difference operator R : L2 (1Rn) --+ L2 (1Rn) by the formula

(2.1) Ru(x) = L ah(x)u(x +h). hEM

Here ah E 0 00 (1R" ), the set M consists of a finite number of vectors h E JRn with integer coordinates.

We introduce the operator RQ = PQRIQ : L 2 (Q) --+ L 2(Q), where IQ: L 2 (Q) --+ L 2 (1Rn) is the operator of extension offunctions from L 2 ( Q) by zero in JRn \ Q, PQ : L2 (1Rn) --+ L2(Q) is the operator of restriction of functions from L2 (1Rn) to Q.

Denote by G the additive group generated by M. Let Qr be the open

connected components of the set Q \ ( U (8Q +h)). hEG

DEFINITION 2.1. A set Qr is called a subdomain. A set R of all subdomains Qr (r = 1, 2, ... ) is called a decomposition of the domain Q.

The decomposition R can be divided into disjoint classes in the following way: Qr1 , Qr, E R belong to the same class if there exists an h E G such that Qr, = Qr, + h. We denote the subdomains Qr by Q,z, where s is the number of class and l is the number of a subdomain in the sth class. Evidently, each class consists of a finite number N = N(s) of subdomains Q,l and N(s) ::=; ([diamQ] +It. A set of classes can be countable.

We define the matrices R, = R,(x) (x E Q,1) of order N(s) x N(s) with the elements

(2.2) rt(x) = { ah(x + h,;) (h = hsj- h,; E M), J 0 (h,i - h,; f/: M),

where h,; is such that Q,; = Q,1 + hsi· Since Q is a bounded domain, by virtue of ( 2.2), the number of different

matrices R, is finite if the coefficients ah are constants. Let n1 denote this number, and let R,. denote all different matrices R, (v = 1, ... , ni).

LEMMA 2 .1. Let the coefficients ah be constants. Then

n1

a(RQ) = U a(R,J, v-:::::::1

where a(RQ) is the spectrum of RQ.

For a proof, see Lemma 8. 7 in [6], Section 8.

THE MIXED BOUNDARY VALUE PROBLEM 409

2. We introduce the set lC by the formula

(2.3) lC = U {Q n (BQ + hl) n [(BQ + hz) \ (8Q + hl)]}, h1,h2EG

Assume that the following condition holds:

CoNDITION 2.2. JJ.n-l (!C n BQ) = 0 and K c !C. Denote by I'v the components of the set BQ \ !C, which are open and

connected in the topology of BQ. By virtue of Condition 2.2, r P E coo. LEMMA 2.2. If (I'v +h) nQ of 0 for 80me hE G, then either I'v +h C

Q, or there is r,. c 8Q \ lC such that I'p + h = r,..

For a proof, see Lemma 7.5 in [6], Section 7.

By virtue of Lemma 2.2, we can decompose the set {l'v + h: fv + h C Q, p = 1, 2, ... ; h E G} into classes in the following manner. The sets r p, + h1

and r v2 + h2 belong to the same class if 1) there exists an h E G such that I'p, + h1 = I'p, + h2 + h, and 2) in the case I'p, + h1 , I'p, + hz c BQ, the directions of the inner normals to 8Q at the points X E f Pl + h1 and X - h E r P2 + hz coincide. Clearly, a set r p c aQ can be in only one class, and a set I'p + h C Q is in at most two classes. We denote a set I'p + h by r ,.1, where r is the number of the class and j is the number of an element in a given class (1 :::; j :::; J = J(r)). Without loss of generality, we shall suppose that

rT!, ... ' I'rJo c Q, I'r,Jo+l> ... 'r,.j c aQ (0:::; Jo = Jo(r) < J(r)).

LEMMA 2.3. For every r = 1, 2, ... , there eXi8ts a unique s = s(r) such that N(s) = J(r) and after some renumbering I',.t c 8Qs1 (l = 1, ... , N(s)).

For a proof, see Lemma 7. 7 in [6], Section 7.

EXAMPLE 2.1. We consider an operator R: L2 (JR:n) -t L2 (JRn) (n ;:o: 2) defined by

(2.4) k

Ru(x) = L a1u(x1 + j, xz, ... , Xn)·

j=-k

Let Q = (0, d) x D, where k is a natural number, d = k + IJ, 0 < IJ :::; 1, D E JR:n-t is a bounded domain (with boundary 8D E coo if n ;:o: 3), a1 E IC.

a) Let B = 1. Then the decomposition n consists of one class of subdomains: Qsl = (l - 1, l) x D (l = 1, ... , k + 1). Moreover, hu = (l - 1, 0, ... , 0) (l = 1, ... , k + 1), and

(2.5) (

ao a1

R _ a-1 ao

1-

a_k a-k+l

... ak )

::: ~k-~1 .

. . . ao

If n :::: 3, then there are three classes of sets rrl: 1) rl,k+l = {0} X D, ru = {l} X D (l = 1, ... ,k), 2) r21 = {l} X D (l = 1, ... ,k+ 1), 3) rsl = (l - 1, l) X fJD (l = 1, ... 'k + 1). Clearly, Jo(1) = Jo(2) = k and J0 (3) = 0. Similarly, if n = 2, we have four classes of sets rrl (instead of the

k+l third class we obtain two classes). For any n;::: 2, JC = U ( {l} x fJD).

l=O

b) Let B < 1. Then the decomposition n consists of two classes of subdomains: Q11 = (l-1, l-1 +B) x D (l = 1, ... , k+ 1) and Q21 = (l-l+B, l) x D (l = 1, ... ,k). Moreover, h11 = (l- 1,0, ... ,0) (l = 1, ... ,k + 1) and h11 = (l-1, 0, ... , 0) (l = 1, ... , k). The matrix R1 of order (k+ 1) x (k+ 1) is given by (2.5). The matrix R2 of order k x k is obtained from R1 by deleting the last row and the last column.

If n :::: 3, then there are four classes of sets rrl: 1) rl,k+l = {0} X D, rll = {l} x G (l = 1, ... , k), 2) T21 = {l- 1 + B} x D (l = 1, ... , k + 1), 3) rsl = (l- 1, l- 1 +B) X D (l = 1, ... 'k + 1), 4) r4l = (l- 1 + B, l) X D (l = 1, ... , k). Clearly, Jo(1) = J 0 (2) = k and Jo(3) = Jo(4) = 0. If n = 2, then we have six classes of sets r rl·

k+l For any n :2:2, K = U [({l-1} x fJD) U ({l 1+ B} x fJD)].

l=l

DEFINITION 2.2. A function <p E C(Q) is said to be M-periodic in Q if <p(x) = <p(x +h) for all x,x +hE Q and hE M.

LEMMA 2.4. Let a junction <p(x) be M -periodic in Q.

Then RQ('Pu) = <pRQu for all u E L2(Q).

For a proof, see Lemma 8.10 in [6], Section 8.

3. We denote by W2'(Q) the Sobolev space of complex-valued functions with the norm

{ }

1/2

llullwf(Q) = L j ID"u(xWdx , JaJ:Sk Q

where a:= (a:!, ... ,an), lo:l = 0:1 + ... + O:n, and 1)0' = Vf' ... TJ~n, TJj = -i 8~ . We denote by W~-112 (f) the space of traces on r with the norm

J

llvllw~-'l'(r) = inf llullw~(Q) (u E Wf(Q): ulr = v), where r C Q is a smooth (n- 1)-dimensional manifold. Various ways of introducing equivalent norms in Sobolev spaces of noninteger order can be found in [7], Chapter 1, for example. Denote by Wf(Q) the closure of the linear manifold C00 (Q) (of compactly supported functions infinitely differentiable in Q) in the space Wf(Q).

By Condition 2.1, the boundary EJQ is Lipschitz. Therefore, for the domains we are considering, the theorem on the continuation of functions in Sobolev space (see Theorem 5 in [8], Chapter 6, §3) is true. As is known, by virtue of this theorem it is sufficient to prove many properties of Sobolev spaces for a bounded domain with smooth boundary. We shall therefore use the results without mentioning each time the conditions on the boundary EJQ. We note that the property that characterizes the space Wf(Q) is not a direct consequence of the theorem on continuation of functions. To complete the picture, we give the proof of this property.

LEMMA 2.5. Wf(Q) = Wfv(Q), where Wfv(Q) = {u E Wf(Q) : TJt- 1ulaQ\K = 0, p, = 1, ... , k}, Vv = -if.Jjf.Jv, a~d v is the unit external normal to f.JQ at the point x E EJQ \ K.

Proof. The inclusion Wf(Q) c Wfv(Q) is obvious. Let us prove the , reverse inclusion.

We introduce the space H&(Q) as a completion of the set C0 (Q \ K) with respect to the norm

where C0 (Q \ K) is the set of infinitely differentiable functions in Q with compact support belonging to Q\K; p E C 00 (Rn\K) is a real-valued function which in some €-neighbourhood of the set K coincides with distance to the set K, and p(x) 2 c > 0 outside this €-neighbourhood.

We define the function 7Jo E C 00 (Rn) as follows: 7J0(x) = 0 (x E K 8 = {x ERn : p(x, K) < 6} ), 7J0(x) = 1 (x f/: K 28

), 0 :S 7J0(x) :S 1 and IIJ!37J0(x)l :S k06-lill (x E Rn; 1;31 :S k), where 0 < 26 < c:. Then for each function wE H~(Q) we have

llw -ry•wll2 k < k! ~ { /(lfll+l,l-kJIV'wi21V"(1 -ry.Wdx < Ho(Q) - L.- JR 26 -

I"YI~k.IPI~k-hl K nq

:::: k2 L r /(hl-k)IV'wi2dx. I"YI~k}K"nQ

Consequently, iiw-ryawiiH8(Q) --+ 0 as o--+ 0. Obviously, the space Hi(Q) is continuously embedded in W~(Q). On the other hand, by the embedding theorems for weighted spaces (see [9], §4 and [10], §10) the space WMQ) is continuously embedded in Hi(Q). Thus, for any function u E W~(Q) we have

(2.6)

By construction, (ryau)(x) = 0 (x E K 6 n Q). We can therefore assume that 1]0U E WMO), supp(ryau) c !1, where !1 c Q, all E C"". By Theorem 11.5 from [7], Chapter 1, § 11, on the characteristic property of the space W~ ( !1) for domains is a smooth boundary, for any o > 0 there is a sequence of functions u, E C""(Q), suppu, C !1 such that

(2.7) ll7!aU- u,llw~(Q) --+ 0 as s--+ 0.

By (2.6) and (2.7), u E W~(Q). D

LEMMA 2.6. Let the coefficients ah of difference operator R be con-stants.

Then for all u E W~(Q)

V" Rqu = RqV"u (Ia I :S: k).

For a proof, see Lemma 8.14 in [6], Section 8. We assume that the following condition is fulfilled:

CONDITION 2.3. For every subdomain Q,1 (s = 1, 2, ... ; l = 1, ... , N(s)) and for each s > 0, there exists an open set G,t c Q,t with boundary aG,t E C1 such that JL,.(Q,1 \ G,t) < s and f.Ln-I(aG,u:::,.OQ,t) < s.

3. Parabolic Differential-Difference Equations. 1. We introduce the unbounded differential-difference operator AR : V(AR) C L 2 (Q) --+ L2(Q) acting in the space of distributions V'(Q) by the formulas

n n

ARu =- L(R;jQUx;)x, + LR;qux, + Roqu (u E V(AR)), i,j=l i:::::l

(3.1)

R.;,ju(x) = L aijh(x)u(x +h) (i,j = 1, ... , n), hEM

R.;,u(x) = L aih(x)u(x +h) (i = 0, 1, ... , n), hEM

M c JR.n is a finite number of vectors with integer coordinates, aijh, aih E coo(JR.n).

DEFINITION 3.1. The operator An is said to be strongly elliptic if there exist constants c1 = c1 (An) > 0 and Cz = c2(An) :::: 0 such that for all u E C00 (Q)

(3.2)

In order to formulate necessary and sufficient conditions of strong ellipticity in an algebraic form, we introduce some notation. Let x E Q,1 be an arbitrary point. Consider all points x1 E Q such that x1 - x E M. Since the domain Q is bounded, the set { x1} consists of a finite number of points I= I(s, x) (I:::: N(s)). We shall number the points x1 so that x1 = x+h,1 for l = 1, ... , N = N(s), x 1 = x, where h,l satisfies the condition Q,l = Q,1 +hsl· We introduce the I xI matrices Aijs(x) with elements a;nx) by the formula

(3.3)

We define the N x N matrix R.;,j 8 (x) so that, if N < I, then Rij,(x) is obtained from Aijs(x) by deleting the last I·-N rows and columns. If N =I, then Rijs(x) is equal to Aijs(x).

By virtue of Theorem 9.2 in [6], Section 9, if for all s = 1, 2, ... , x E Q,1 ,

and 0 # ~ E JR.n the matrices

n

L (Aijs(x) + Aij,(.r))~i~j id::::l

are positive definite, then the operator An is strongly elliptic. On the other hand, if the operator An is strongly elliptic, then by Theorem 9.1 in [6],

Section 9, the matrices

n

L (R;js(x) + R'fjs(x))~i~j i,j=l

are positive definite for all s = 1, 2, ... , x E Q81 , and 0 ojo ~ E lRn.

REMARK 3.1. We consider the· operator AR = ARQ, where RQ n

PQRIQ, Ru(x) = I; ahu(x + h), ah E JR, Av = - I; 8~ aij 8~v, aij = hEM i,j=l t J

aii E C00 (lRn) are real-valued M-periodic functions in Q. n

Let I; aij(x)~i~j > 0 for all x E Q.1, s = 1, 2, ... , and 0 o/o ~ E lRn. i,j=l

Then the operator AR is strongly elliptic iff the matrices Rs + R; are positive definite for all s = 1, 2, ... (see Example 9.3 in [6], Section 9). Moreover, in this case c2(AR) = 0.

2. We consider the differential-difference equation

(3.4) u1(x, t) + ARu(x, t) = j(x, t) ( (x, t) E Qr)

with boundary condition

(3.5) ulrr = 0 ((x,t) Err)

and initial condition

(3.6) ult=O = tp(x) (x E Q),

where Qr = Q X (0, T), rr = aQ X (0, T), 0 < T < oo, f E L2(Qr), and 'P E L2(Q).

Further we shall suppose that the operator AR is strongly elliptic. In this case problem (3.4)-(3.6) is called the first mixed problem for parabolic differential-difference equation. Without loss of generality, we can assume that in inequality (3.2) c2 = 0.

Let A : D(A) C L2 (Q) --+ L2 (Q) be a closed operator, and let D(A) be dense in L 2 (Q). We introduce the Hilbert spaces D(A) and V(A) -L2(0, T; D(A)) with the inner products

(tp, 1/J)v(A) = (Atp, A'I/J)L,(Q) + (tp, 1/J)L,(Q) (tp, 1/J E D(A)),

T

( u, v )v(A) = j ( u, v )v(A)dt. 0

THE MIXED BOUNDARY VALUE PROBLEM

We also define the Hilbert space W(A) = {w E L2 (0,T;D(A)) L2(0, T; L2( Q))}

T

(u, v)w(A) = (u, v)v(A) + J (88

U, 8

8 v) dt.

0 t t Lz(Q)

Here we consider derivatives in the sense of distributions in Qr.

415

DEFINITION 3.2. A function u E W(AR) satisfying (3.4), (3.6) is called a strong solution of problem (3.4)-(3.6).

To prove an existence of strong solutions, we make use of semigroup theory.

DEFINITION 3.3. A strongly continuous semigroup of operators {T1}

(t 2': 0) in a Hilbert space His said to be contractive if IIT1II ::; 1 (t 2': 0).

DEFINITION 3.4. A strongly continuous semigroup of operators {T1}

(t 2': 0) in a Hilbert space H is said to be stable if lim IIT110II = 0 for every l-400

<p E H.

Denote Ll.w = {z E C: iargzl < w}, where 0 < w.

DEFINITION 3.5. A family of linear bounded operators {Tz} (z E Ll.w) in H is called analytic semigroup in Ll.w if 1) the function z -+ Tz is analytic in Ll.w, 2)To =land lim Tzx =X (x E H),3) Tz,+zz =Tz,Tz, (z1,Z2 E Ll.w)·

z-+0,zE.6.w

A semigroup T1 is said to be analytic if there is an analytic extension Tz of the operator-function T1 into some angle Ll.w-

THEOREM 3.1. Let the domain Q satisfy Condition 2.1. Assume that the operator AR is strongly elliptic, and that c2 (AR) = 0.

Then - AR is an infinitesimal generator of analytic contractive semigroup {T1} (t 2': 0) in L2 (Q). Moreover, the semigroup {T1} (t 2': 0) in L2 (Q) is stable.

Proof. From Theorems 3.1 and 3.2 in [1] it follows that -AR is a generator of analytic contractive semigroup {Tt} (t 2': 0). On the other hand, Theorem 10.1 from [6], Section 10 implies that the spectrum a-( -AR) is discrete and a-( -AR) c {.\ E C: Re .\ < 0}. Therefore, by virtue of Stability Theorem in [11], the semigroup {T1} (t 2': 0) is stable. D

Let H 1 and H be Hilbert spaces such that H 1 is dense in H with continuous injection H 1 c H. For every 7/J E H and for all t > 0, we define the functional

We now introduce the interpolation space 00

[H1,H]I;2 = {'1/> E H: J r 2K 2(t,'lj>;H1,H)dt < oo} 0

with the norm 00

II'I/>II[H,Hh;2 = (11'1/>11~ + J r 2 K2(t, '1/>; H1, H)dt?1

2.

0

LEMMA 3.1. Let the domain Q satisfy Condition 2.1. [ '2 l '1( Then W2 (Q), L2(Q) 112 = W2 Q).

Proof. The proof is similar to the proof of Theorem 11.6 in [7], Chapter 1 and is based on Lemma 11.3. in [7], Chapter 1. We should only use the Calderon method of extension for functions from Lipschitz domain instead of the Hestenes method, see Theorem 5 in [8], Chapter 6, §3. D

LEMMA 3.2. Let the domain Q satisfy Condition 2.1. Then W:f(Q) is continuously imbedded into V(AR)·

Proof. From Lemma 8.13 in [6], Section 8, it follows that the operator R;iQI)~; is mapping continuously W:f(Q) into W:f(Q). Therefore W:f(Q) C

D(AR) and the operator AR is mapping continuously W:f(Q) into L2(Q). D

LEMMA 3.3. Let the domain Q satisfy Condition 2.1. Assume also that the operator AR is strongly elliptic, and that c2(AR) = 0.

Then problem (3.4)- (3.6) has a unique strong solution iff <p E [V(AR), L2(Q)h/2· Moreover-, this solution is given by the for-mula

(3.7)

t

u(x, t) = Tt<p(x) + J Tt-sf(x, s)ds,

0

where {7t} (t 2 0) is the analytic semigroup generated by the operator -AR.

Proof. We shall consider problem (3.4)-(3.6) as an abstract Cauchy problem for parabolic equation in the space L 2 (Q). By virtue of Theorem 3.7 in [12], Chapter 1, problem (3.4)-(3.6) has a unique strong solution iff the following inequality holds:

T

(3.8) J IIARTt'PIIL(Q)dt < oo. 0

Moreover, this solution is given by (3.7). By Theorem 3.1, the semigroup is analytic and contractive. Therefore Theorem 1.14.5 in [13], Chapter 1 implies that inequality (3.8) takes place iff <p E [:D(AR), L2 (Q)h;2 • 0

THEOREM 3.2. Let the domain Q satisfy Condition 2.1. We also as-sume that the operator AR is strongly elliptic, and that c2 (AR) = 0.

Then for all f E L2 (Q) and <p E W}(Q) problem (3.4) - (3.6) has a unique strong solution. Moreover, this solution is given by formula (3.7).

Proof. By virtue of Lemma 3.3, it is sufficient to prove that WJ(Q) c [:D(AR), Lz(Q)h;z.

Let 0. Therefore if r 1K(t,<piWi(Q),L2 (Q)) E L2 (0,oo), then r 1 K(t, 'Pi :D(AR), L2 (Q)) E Lz(O, oo). Hence [Wi(Q), L2 (Q)h;z C [:D(AR), Lz(Q)h;z. Thus, by virtue of Lemma 3.1, <p E W}(Q) = [W:f(Q), Lz(Q)h;2 c [:D(AR), Lz(Q)h;z. 0

4. Space of Initial Data. 1. By virtue of Lemma 3.3, it is natural to consider the space [:D(AR), L2 (Q)h;2 as the space of initial data for problem (3.4)-(3.6). From Theorem 3.2 it follows that WJ(Q) C [:D(AR), Lz(Q)h;2 •

In this Section we give sufficient conditions of coincidence for these spaces.

THEOREM 4.1. Let assumptions of Theorem 3.1 hold, and let AR : :D(AR) C Lz( Q) -+ L2 ( Q) be a selfadjoint operator.

(

0 1 Then [:DAR), Lz(Q)h;z = W2 (Q).

Proof. The operator An is selfadjoint and positive. Therefore there exists a selfadjoint operator A~2 • Hence, using inequality (3.2) and boundedness of difference operators in the space L 2 (Q), we obtain

clllull~i(Q) :S: (Anu, u)L,(Q) = (A~2 u, A~2u)L,(Q) =

n n

= L (R;jQUx;, UxJL,(Q) + L(R;QUx, u)L,(Q)+ i,j=l i=l

( 4.1)

Since the operator A~2 : :D(A~2 ) C L2 (Q) -+ L2 (Q) is closed, from

(4.1) it follows that :D(A~2) = W:/(Q). By virtue of Theorem 1.18.10 in [13], 1/2 ° Chapter 1, we have [:D(An), Lz(Q)h;z = :D(AR ) = W:f(Q). 0

THEOREM 4.2. Let assumptions of Theorem 3.1 hold. Let there exist a selfadjoint strongly elliptic differential-difference operator BR of form (3.1) such that c2(BR) = 0 and :D(AR) is continuously imbedded into :D(BR)·

Then [:D(AR),L2(Q)h12 = WJ(Q).

Proof. By Theorem 3.2, WJ(Q) C [:D(AR), L2(Q)h;2. We prove the inverse inclusion.

Theorem 4.1 implies that [:D(BR), L2(Q)h;2 = WJ(Q). From conditions of the theorem we obtain

K(t, ({); :D(BR), L2(Q)) :::; cK(t, ({); :D(AR), L2(Q)),

where c > 0. Therefore [:D(AR),L2(Q)hJ2 C [:D(BR),L2(Q)hJ2 = WJ(Q). D

2. Unlike elliptic differential equations, the smoothness of solutions to equation

(4.2)

' can be violated inside domain Q even for f E coo ( Q), see [ 6], Sections 11 and 12. This means that :D(AR) # WJ(Q) n Wi(Q). However, by virtue of Theorem 11.2 in (6], Section 11, if u E :D(AR) is a solution of equation (4.2) for f E L2(Q), then u E Wi(Q,t \ K!') for every c > 0 (s = 1,2, ... ;1 = 1, ... ,N(s)), where}("= {x E llitn: p(x,JC) < c}. Under additional assumptions on smoothness near the set JC, we prove that [:D(AR),L2(Q)hJ2 = WJ(Q). D

THEOREM 4.3. Let Conditions 2.1 - 2.3 hold. Let the numbers S0

and r0 of differ·ent classes of subdomains Q,t and different classes of surfaces ' frm be finite, and let each subdomain Q,t (s = 1, ... , S0 ; 1 = 1, ... , N(s)) be

Lipschitz. Assume also that the operator AR is strongly elliptic, c2(AR) = 0, and that every solution u E :D(AR) of equation (4.2) belongs to Wi(Q,t) (s= 1, ... ,S0;l= 1, ... ,N(s)).

Then [:D(AR),L2(Q)hJ2 = W21(Q).

Proof. 1. First we construct an equivalent norm in the space :D(AR)· Denote by w;•n(Q) the space of functions u such that u E W~(Q,t)

(s = 1, ... , So; l = 1, ... , N(s)) with the norm

(4.3)


Clearly, W{(Q) c w;•w(Q) and

( 4.4)

We now describe the space D(An). From condition of the theorem it follows that D(An) c Wi(Q) n w;•n(Q). Let u E Wd(Q) n w;•n(Q). Then Rqu E w;•n(Q). By virtue of Condition 2.3, integrating by parts over 0,1

and passing to a limit as E: ~ 0, we have

J LR.jQUx/iix,dx = -.! L(R.jQUx;Jxivdx+ Qsl t,J Qsz Z,J

(4.5) + / L(R.jQUx; )vlaq,,vc cos(v, xi)dx'

8Q,,\IC I,J

for all v E C00 (Q), where vis the unit external normal vector to EiQsl at the point x' E 8Q,1 \ K.

Summing overs, l, from (4.5) we obtain (4.6)

/I:: R;jQUx/Vx,dx = /tvdx + L .! L(R.jqux;)vlaq,,\K cos(v, xi)dx'. Q z,J Q s/ 8Qst\K t,J

Here J E L2(Q) is given by the formula f(x) =- 'f:(R;jqUx;)x,(x) for x E I,J

Qsl· From ( 4.6) it follows that u E D(An) if and only if

(4.7) L .! L(R.jQUx;)vlaq,,\Kcos(v,xi)dx'=O. s,l &Qst\K id

Each set 8Q,1 \ K consists of finite number of smooth surfaces frm· By virtue of Lemma 2.3, for every fixed r there exists a unique s = s(r) such that N(s) = J(r) and after some renumbering fr 1 c 8Q,1 (l = 1, ... , N(s)). Clearly, there exist p = p(r) and m = m(r) such that rr1 C EiQpm and Qprn # Q,1 . We also renumber subdomains of the pth class so that fr1 c 8Qsl n EiQpl n Q (l = 1, ... , Jo).

Then ( 4. 7) will take the form

(4.8) L(R.jQUx;)szlr, cos(vr~,x;) = L(R.jQUxJvzlr, cos(vrl,xi)

(r = 1, ... ,r0 : r E {r: J0 (r) > O};l = 1, . .. ,J0 ).

Here Ost is a restriction of function to Q81, Vrl is a unit normal to rrl at a point X E rrl·

We can rewrite (4.8) as following:

(4.9)

N(p)

N(s)

L L rlf•(x)(ust)x1 lr.,+h,.-h,, cos(Vrt,x;) = i,j t==l

= LL rl~(x)(upm)x; lr,,+hpm-hp, cos(vrl, X;) (r = 1, ... , ro; l = 1, ... , lo)· i,j m=l

Here r;f•(x) = aijh(x) if h = hst- hst E M, r;f•(x) = 0 if h = hst- hst ~ M ( cf.(2.2) ).

Therefore a function u E Wi(Q) n Wi'n(Q) belongs to 'D(AR) iff equalities (4.9) hold. Conditions (4.9) define a closed linear subspace in Wi(Q) n Wi'n(Q). We denote this subspace by W~'n(Q). Since 0 ~ O"(AR) and the operator AR : w~·n(Q) -+ L2(Q) is bounded, the Banach inverse operator theorem implies that the norms llullv(AR) and ll·ullw£·"(Q) are equivalent in 'D(AR)·

2. Now we prove that ['D(AR),L2(Q)h12 = Wi(Q). Since subdomains Q81 are Lipschitz, from interpolation theorems 7.1, 9.1,

and 9.2 in [7], Chapter 1 it follows that [W:r(Q.t),L2(Q.t)h/2 = Wi(Q.z). By assumption of the theorem, the number S0 is finite. Therefore

(4.10)

On the other hand, from the first part of the proof it follows that the imbedding 'D(AR) c Wi'n(Q) is continuous. Hence

( 4.11)

Let 'P E ['D(AR), L2(Q)]I/2. Then, by Theorem 1.6.2 from [13], Chapter 1, for every m > 0 there exists 'Ptm E 'D(AR) such that

(4.12) II'P- 'PtmlliV(AR),Lz(Q)h;2 < 1/m.

From (4.10), (4.11), and (4.12) it follows that

( 4.13) II'P- 'Ptmllwi•"(Q) < cjm,

where c > 0 does not depend on m.

Since D(AR) C Wi(Q), by virtue of (4.4), for every m > 0 there exists 'P2m E C00 (Q) such that

(4.14) [['P!m- 'P2mllwi·"'(Q) = [['Plm- 'P2m[[wJ(Q) < 1/m.

Hence

Therefore <p2m ...-t <pin the space Wi•n( Q) as m ...-t oo. Thus { 'P2m} is the Cauchy sequence in Wi•n(Q). By virtue of (4.4) {'P2m} is also the Cauchy sequence in Wi(Q). From uniqueness of limit it follows that <p E Wi(Q). D

EXAMPLE 4.1. Let n = 1, and let Q = (O,d), where d = k + (), 0 < () :S 1, k is a natural number. We consider the differential-difference operator given by

(u E D(AR) = {u E Wi(O, d): Rqu E W:f(O, d)}).

k Here Rq = PqRiq, R0q = PqR0Iq, Ru(x) = 2:: aiu(x + i), Rou(x) -

k

2:: aoi11(x+i), ai,aoi E C. i=-k

i=-k

Lemma 2.6 implies that (Rqu)" = (Rqu')' for u E D(AR). Hence the operator AR can be represented in form (3.1).

If()= 1, then the decomposition R for the interval (0, d) consists of one class of subintervals: Q 11 = ( l - 1, l) (l = 1, ... , k + 1). If () < 1, then the decomposition R consists of two classes of subintervals: Qu = ( l-1, l-1 + 0) (l = 1, ... 'k + 1) and Q2l = (l- 1 + e, l) (l = 1, ... 'k).

We define the matrices R1 and R01 of order ( k + 1) x ( k + 1) with the elements

(4.15) r}j = aj-i, r~f = ao.j-i (i,j = 1, ... , k + 1),

respectively (cf. (2.2)). If()< 1, we also consider the matrices R2 and R02 ,

which are obtained from R1 and R01 , respectively, by deleting the last column and the last line.

Assume that the matrix R1 + Ri is positive, and that the matrix R01 + R01 is nonnegative. Then the matrix R2 + R; is positive definite, and the matrix

~2+R(j2 is nonnegative. Therefore, by virtue of Remark 3.1 and Lemma 2.1, the operator AR is strongly elliptic and c2(AR) = 0. On the other hand, by Theorem 3.2 in [6], Section 3 on the smoothness of generalized solutions for differential-difference equations, if u E V(AR), then u E Wi(Q,1). Thus, by virtue of Theorem 4.3 and Lemma 3.3, problem (3.4)-(3.6) has a unique strong solution in the rectangle Qr = (0, d) x (0, T) iff <p E Wi(O, d).

EXAMPLE 4.2 (SEE EXAMPLE 2.1). Let Q = (O,d) x D, where DC JRn-l (n ::::_ 2) is a bounded domain (with boundary oD E coo if n ::::_ 3), d = k + B, 0 < B ::; 1, k is a natural number. We consider the differentialdifference operator AR given by

k

Here RQ = PQRIQ, ~Q = PQRolQ, Ru(x) = I: aiu(x1 +i,x2, ... ,xn), i;;;;;-k

k n ai E CC, ~u(x) =I:aoiu(xl+i,x2,···,xn), aoi E CC, Av(x) =-I: a~aij(x)x

i=-k i,j-=1 t

l!~j v(x), aij = aji E C00 (JR") are real-valued 1-periodic functions. n

Lemma 2.6 implies that ARou =- I: a8 R.;3·Qal!· u (u E V(AR)), where . . I x~ x3 2,)=

ll.;jQV = PQ!l.;jlQ, R.;j = aijR· Hence the operator AR can be represented in form (3.1).

If e = 1, then the decomposition n consists of one class of subdomains: Qll = (l- 1, l) X D (l = 1, ... , k + 1). If e < 1, then the decomposition n consists of two classes of subdomains: Q11 = (l- 1,1- 1 +B) x D (l = 1, ... ,k+ 1) and Q2t = (l-1 +B,l) x D (l = 1, ... ,k).

We define the matrices R1 and R01 by formula (4.15). We also consider the matrices R2 and Ro2 similarly to Example 4.1.

Let I: aij(x).;i.;j > 0 for all X E Q,] (s = 1, 2 if e < 1 and 8 = 1 if e = 1) i,j

and 0 i' .; E JRn. Assume also that the matrix R1 + RI is positive definite, and that the matrix ~1 + R01 is nonnegative. Then the matrix R2 + R?. is positive definite, and the matrix R02 + R(J2 is nonnegative. Therefore, by virtue of Remark 3.1 and Lemma 2.1, the operator AR is strongly elliptic and c2(AR) = 0. On the other hand, by Theorem 23.2 in [6], Section 23 on the smoothness of generalized solutions for elliptic differential-difference equations in a cylinder, if u E V(AR), then u E Wi(Qst)· Thus, by virtue

of Theorem 4.3 and Lemma 3.3, problem (3.4)-(3.6) has a unique strong solution in Qy = Q x (0, T) iff <p E Wi(Q).

We consider a particular but very important case of Example 4.2.

EXAMPLE 4.3. Let Q = (0, 2) x (0, 1). We study the differential-difference operator AR given by

Here Rq = PqRiq, Ru(x) = u(x1 , xz) + a1u(x1 + 1, xz) + a_Ju(xl- 1, xz). The decomposition R consists of two subdomains Q11 = (0, 1) x (0, 1)

and Q21 = (1, 2) x (0, 1). The matrix R1 = ( 1 ~1 ) .

a-1 Assume that la1 + a_ 11 < 2, i.e. the matrix R1 + Rt is positive definite.

Then from Example 4.2 it follows that problem (3.4)-(3.6) has a unique strong solution in Qy iff <p E W](Q).

EXAMPLE 4.4. Consider the operator AR given by

(u E D(AR) = {u E Wi(Q): ARu E Lz(Q)}).

Here R;_q = PqR;_Iq (i = 1,2), R1u(x) = 2u(x1 ,x2 ) + u(x1,x2 + 1)+ +u(xb Xz - 1), Rzu(x) = 2u(x1 , xz) + u(x1 + 1, Xz) + u(x1 - 1, xz), Q = (0, 2) X (0, 2).

The decomposition R consists of four subdomains Q11 = (0, 1) x (0, 1), Q12 = (1, 2) x (0, 1), Q13 = (0, 1) x (1, 2), Q14 = (1, 2) x (1, 2). From the definition of the set K (see (2.3)) we have K = { ( i, j) : i, .i = 0, 1, 2}. Clearly,

the matrices R 11 = R21 = ( ~ ~ ) . Hence, by Lemma 2.1, the selfadjoint

operators Riq : L2 (Q) -r L2 (Q) are positive definite. Thus it is easy to see that the operator AR is strongly elliptic and cz(AR) = 0. By virtue of Theorem 11.2 in [6], Section 11, if u E D(AR.), then u E Wi(Q 11 \ K') (l = 1, ... , 4) for every E: > 0. Moreover, in this case it was proved that D(AR) =

WJ(Q) n Wi(Q) (see Example 12.2 in [6], Section 12). Therefore Theorem 4.3 and Lemma 3.3 imply that problem (3.4)--(3.6) has unique strong solution in Qy iff 'P E W](Q). The same statment follows from Theorem 4.1, since the operator AR is selfadjoint (see Theorem 10.2 in [6], Section 10).

REMARK 4.1. In [14] equation (4.2) was considered in an arbitrary plane domain. A difference operator could have shifts in different directions.

It were stated sufficient conditions for smoothness of solutions near the set K in subdomains Qsl· However, there are examples when smoothness of solutions of equation (4.2) is violated near the set K (see Example 11.2 in (6], Section 11). It would be very interesting to describe the space of initial data [D(AR), L2(Q)h;2 in this case.

REFERENCES

[1 J A.L. Skubachevskii and R. V. Shamin, The first mixed problem for parabolic differential-difference equation, Mat. Zametki, 66 (1999), 145-153 (in Russian).

[2] M.A. Vorontsov, N.G. Iroshnikov, and R.L.Abernathy, Diffractive patterns in a nonlinear optical two-dimensional feedback system with field rotation, Chaos, Solitons, and Fractals, 4 (1994), 1701-1716.

[3] A.V. Razgulin, Rotational multi-petal waves in optical system with 2-D feedback, Chaos in Optics, Proceedings SPIE, 2039 (1993), 342-352.

[4] A.L. Skubachevskii, On the Hopf bifurcation for quasilinear parabolic functional differential equation, Differentsial'nye Uravneniya, 34 (1998), 1394-1401 (in Russian).

[5] A.V. Bitsadze and A.A. Samarskii, On some simple generalizations of linear elliptic boundary value problems, Soviet Acad. Sci. Docl. Math., 185 (1969), 739-740 (in Russian).

[6] A.L. Skubachevskii, Elliptic functional differential equations and applications, Basel-Boston-Berlin, Birkhiiuser, 1997.

[7] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, vol.1, Springer-Verlag, New York-Heidelberg-Berlin, 1972.

[8] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Series, No 30, Princeton Univ. Press, Princeton, 1972.

[9] V.A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or corner points, Trudy Moskov. Mat. Obshch., 16 (1967), 209-292 (in Russian).

[10] V.G. Maz'ya and B.A. Plamenevskii, Lp-<Jstimates of solutions of elliptic boundary value problems in domains with ribs, Trudy Moskov. Mat. Obshch., 37 (1978), 49-93 (in Russian).

[11] W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroup, Trans. Amer. Math. Soc., 306 (1988), 837-852.

[12] A. Ashyralyev and P.E. Sobolevskii, Well-posedness of parabolic difference equations, Basel-Boston-Berlin, Birkhiiuser, 1994.

[13] H. Triebel, Interpolation theory, function spaces, differential operators, NorthHolland, Amsterdam-New York-Oxford, 1978.

[14] A.L. Skubachevskii, Elliptic problems with nonlocal conditions near the boundary, Mat. Sb., 129 (171) (1986), 279-302 (in Russian).

VOLUME 8 2001, NO 3-4 PP. 425-434

ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

S.A. STEPIN '

Abstract. This paper deals with two problems concerning the asymptotic behavior of solutions to second order ordinary differential equations near singular points and at infinity. The first problem consists in determination of branching type for solutions to the linear differential equation by its coefficients. The second problem concerns the asymptotic equivalence for solutions to quasilinear differential equation and corresponding linear one.

1. It is assumed here that the singular point z = 0 of the equation

(1) dw

za(z) dz + b(z)w = 0,

is regular, i.e. coefficients a(z) and b(z) are analytic in a neighbourhood of zero with at least one of the values a(O), b(O), b'(O) is different from 0.

Let >. 1 and >.2 be the roots of the characteristic equation

>.2 + (a(O)- 1)>. + b(O) = 0,

Re >. 1 :2: Re >.2 . It is well--known that in the nonresonance case (>.1 - ,\2 tj Z) equation (1) has a fundamental system of solutions of the form

where functions <p(z) and 'ljJ(z) are analytic in some neighbourhood of point z = 0. On the other hand, if >. 1 - ,\2 E Z (resonance case), then the fundamental system of solutions to equation (1) has the form

(2) w1(z) = z>.'<p(z), w2(z) = z>-'1/J(z) + Cw1(z)lnz,

' Moscow State University, Faculty of Mechanics and Mathematics, 119899 Moscow, Russia

425

426 S.A. STEPIN

while coefficient C may "happen to" be equal to zero and hence the fundamental system of solutions looks the same as in the nonresonance case.

The statement and investigation of a question concerning the structure of solutions to ordinary differential equation in a neighbourhood of its regular singularity go back to Frobenius and Fuchs (cf. [1]). Some conditions for the absence of logarithmic term in the fundamental system of solutions can be found in [2]; see also [3].

We give an explicit formula for the coefficient C at the logarithmic term of the solution w2(z) in terms of a(z) and b(z). To formulate this result, let us make a substitution

w(z) = y(z) exp ( -~ j a~) dz)

that changes equation ( 1) to the form

(3) d2y

z2- + q(z)y = 0, dz2

where q(z) = b(z)- a2 (z)/4+ (a(z) -za'(z))/2 is analytic in a neighborhood 00

of zero : q(z) = 2:.:: Qk zk. The structure of fundamental system of solutions k=O

to equation (3) near the point z = 0 is the same as for equation (1) while characteristic exponents 11-1 and 11-2 of transformed equation ( 3) are equal to .\1 + a(0)/2 and .\2 + a(0)/2 respectively. Below the notation Fk = k(k + .\2 - .\1), kEN, is used.

THEOREM 1. If ..\1 - .\2 = n EN then equation (1) has a fundamental system of solutions of form (2), where <p(O) = 'lj;(O) = 1 and

Proof. 1) Let y1(z) be the solution of equation (3) corresponding to characteristic exponent tJ-1 and such that limy1 (z) z-~'1 = 1. The second

Z-)Q

solution to (3) is sought in the form

00

Y2(z) = z~'2 [1 + :2::>kzk] + Cy1(z) lnz. k=l

SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS 427

Substituting this expression into equation (3) and setting the coefficients at zM, zM+l, ... , zi'l equal to zero, one can find ck, k = 1, ... , n - 1, and C successively. Thus we get the system of equations

k-1

Fk ck + 2:: ck-1 q1 + qk = 0 , k = 1, ... , n - 1 , 1=1

n-1

n C + 2:: Cn-1 ql + qn 0 . 1=1

2) By induction in n, we prove the formula

n-1

(4) + 2::(-1lj 2:: j=l kj< ... <kl

Pn-k, qk,-k, · · · qk;

Fk, Fk, ... Fkj

It is immediately seen that this equality holds for n = 3 :

Assuming that the formula ( 4) is true for n = m let us consider the sum

C1Pm + C2Pm-1 + . · · + Cm-1P2 + CmP1 ,

where Cm is determined by the relation

According to inductive assumption we have

C1Pm + C2Pm-1 + · · · + Cm--1P2

m-1

2:: (-llj 2:: j=2 kJ' < ... <kl

and hence,

Pm-k,-1-lqk,-k, · · · qkj Fk, ... Fkj

428

_ _ plqm + Fm

Observing that

+ 2::

S.A. STEPIN

qm-kt qk,-k, · · · qk; + ·Fk, ... Fk;

l:Ski+l <···<k1:Sm-l

Pm-k, +1 qk,-k, · · · qk;+t

Fk, ... Fk;+t

2:: I:Skj+l < ... <kl::_:;m

Pm-kt + 1 qk,-k, • • · qk;+t

Fk, ... Fki+'

2 :<::; j :<::; m - 2 , and grouping terms of equal homogeneity degree in the sum { (clPm + C2Pm-1 + ... + Cm-1P2) + CmP1} one obtains formula (4) for n = m + 1.

3) As a consequence of (4), we have

n-1

+ 2::(-l)j 2:: j:=2 kj< ... <kt

qn-kt qk,-k, · · · qk;

Fk,Fk, ... Fk;

Substitution of this expression into the right-hand side of the equality

leads to the required formula for the coefficient C. Thus, necessary and sufficient condition for the solution y2(z) not to contain the logarithmic term is as follows :

n-1

(5) + 2::(-l)j 2:: j=2 k;< ... <kt

qn-kt qk,-k, · · · qk;

Fk,Fk, ... Fk; - 0.

In this case the solution y2(z) depends on an arbitrary constant en and the corresponding term in y2(z) with Cn as a factor is proportional to y1(z);


without loss of generality (for the construction of fundamental system of solutions) one can put c, = 0 .

The general solution of equation (1) may have at most algebraic singularity at zero if and only if )q, .A2 E IQ, .A1 # .A2 , and, moreover, condition (5) has to be satisfied when .\1 - .\2 = n E Z.

2. In this section the question about asymptotic behavior at infinity of solutions to the equation

(6) y" + Q(x)y + yK = 0 , K, EN ,

with complex-valued coefficient Q(.:c) = p(x) + iq(x) will be studied. For quasilinear equations with real coefficients, conditions are known (see

[4]) that guarantee asymptotic equivalence of solutions to such equations and corresponding linear ones. Under these conditions in the case when q(x) = 0 and p(x) > 0 solutions to (6) possess the following asymptotic behavior as x-+oo:

(7)

(8)

Our goal is to construct solutions to equation ( 6) with prescribed asymptotic behavior (7)-(8) at infinity (similar to Jost solutions in the potential scattering theory). As regards the method we use to construct such solutions, it may be characterized as a non-linear variant of WKB method (see [5]).

THEOREM 2. Let p(x) > 0 for x 2: Xo, p E C 2 [xo, oo), q E C[xo, oo) and assume that

(9)

430 S.A. STEPIN

Then equation (6) has a solution y(x) with asymptotics (7)-(8) for arbitrary A,BEC.

Proof. 1) Equation (6) is equivalent to the system

The change of variables ( cf. [5])

carries this system to the following form

p'(x) ( 1 0 ) ( 1 1 ) } ( u1 ) - 4p(x) 0 1 + a(x) -1 -1 u2

where

·{ 1 p"(x) 5 p'(x)2 } q(x)

a(x) = ~ -Sp3/2(x) + 32p5/2(x) - 2v'P(X}.

In what follows, we use the notations

Making use of the substitution

we obtain the system

v; = a(x)(v1 +exp(-2iS(x))v2) + ,B(x)(v1 +exp(-2iS(x))v2f,

2) The latter one is replaced by the system of integral equations

+ ,6(tl(vl +exp(-2iS(t))v2r} dt,

which will be solved by successive approximations method:

- ,6(t) ( exp(2iS(t))v\n-l) + v~n--l)) n} dt,

v\0)(x) =A, v~0)(x) =B. By condition (9), there exists a 2 x0 such that

[''{la(x)l+l,6(xll}dx < rnin{C-I:I,C-IBI}.

where C > max{IAI, lEI} and K = rnax{2C, (2C)n}. Assuming that lv;n-l)(x)l::; C, j = 1,2, for x 2 a, we get

432 S.A. STEPIN

::; /A/ + K [" { Ja(t)J + /,B(t)/} dt ::; C ;

similarly

3) The inequality

which holds for x ;::: x0 is a starting point for an inductive proof of the following estimate valid under condition x ;::: a

(10) Jv)nl(x)- v)n-l) (x)/ ::; K ~!n e(x)n, j = 1, 2,

where M = max { 2, 2x;(2C)~-l} and

e(x) = ["{la(t)/+/,B(t)/}dt.

In fact for x ;::: a we have

Mn+l 1oo { } Mn+l < K ---n! X e(t)n /a(t)/ + /,B(t)/ dt = K (n +I)! e(x)n+l .

4) By virtue of (10), the series

00

(11) Vj(x) = vj0l(x) + L ( vjn+l)(x)- vjn)(x)), j = 1, 2 n=O

converge absolutely and uniformly on the semiaxis x 2': a and therefore these series represent continuous solutions to the system of integral equations in question. It follows from (10) and (11) that

v1(x) = A + c:1(x), v2(x) = B +s2(x),

lc:j(x)l < K(exp(Mp(x)) -1), j = 1, 2.

Taking into account the formula

y(x) = v·-l/4(x) (exp(iS(x))v1(x) + exp(-iS(x))v2(x))

which gives a solution to equation (6), we find out that

+ p- 114 (x)exp(-i1~ yp(i)dt) [B+c:2(x)],

where c:j(x) = O(p(x)), as x -+ oo, j = 1, 2. Thus the solution y(x) possesses the required asymptotic behaivior (7).

5) For the derivative y'(x) one has the following representation:

as x -+ oo. We shall show that p'(x)p-312(x) -r 0 when x -+ oo; with regard to this fact one gets asymptotics (8).

434 S.A. STEPIN

The existence of finite lim p'(x)p-312(x) = L immediately follows from · x-+oo

the relationship

1" p"(t) p'(x) p'(xo) 31" p'(t)2

~'-;'-,- dt = - + - dt xo p3/2(t) p3/2(x) p3/2(xo) 2 xo p5/2(t)

in view of convergence of integral (9). Since we have

p'(x)2 - p'(x)211" p'(t) - -1/2 ,-1 p5/2(x) - p3(x) xo 2p3/2(t) dt p (xo) ,

the assumption L # 0 leads to the inequality

p'(x)2 C > p5/2(x) - x

which holds for sufficiently large x ;::: x0 and some C > 0. This obviously contradicts condition (9), and thus the proof is complete. D

For the coefficient Q(x) = x' conditions of theorem 2 are satisfied if and only if 1 (~> + 1) > 4. In the case 1 = 1 and K = 3 one obtains the second Painleve equation

y" + xy + y3 = 0

for which asymptotics (7)-(8) do not hold if AB # 0. These examples show that assumptions of theorem 2 are in a sense exact.

REFERENCES

[1] E.L. Ince, Ordinary differential equations, Longmans, Green & Co. Ltd., London, 1927.

[2] W.A. Harris, Logarithmic solutions of linear differential equations with singularity of the first kind, Appl. Anal., 8 (1978), 171-174.

[3] N.G. Grigorenko, Logarithmic singularities of Fuchs type equations and finiteness of monodromy group, Matern. Zametki, 33 (1983), 881-884.

[4] I.T. Kiguradze and T.A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations, Kluwer Academic Publishers, Dodrecht-Boston-London, 1993.

[5] M.V. Fedoryuk, Asymptotic methods for linear ordinary differential equations, Nauka, Moscow, 1983.

VOLUME 8 2001, NO 3-4 PP. 435-446

ON SPECTRAL PROBLEMS ARIZING IN THE THEORY OF FUNCTIONAL DIFFERENTIAL EQUATIONS*

V. V. VLASOV t

Introduction. In this paper we present results about asymptotic behavior and estimates of the strong solutions of differential-difference equations of neutral type and study certain spectral questions concerning Riesz basisness of the system of the exponential solutions of the above mentioned equations in Sobolev spaces on the interval of delay.

The paper consists of three parts. We present the statements of the results in the first part, the proofs of the main statements -. in the second one, and remarks and comments - in the third part.

1. Definitions, Notation, and Statements of the Results. Consider the following initial value problem for a differential-difference equation:

(1) tE~,

(2) u(t)=y(t), tE[-h,O), u(+O)=<po=v(-0).

Here B1, D1 (j = 0, 1, ... , n) are matrices (m x m) with constant complex elements; the real numbers hJ are such that 0 = h0 < h1 < ... < hn = h, vector-valued function y(t) E Wi(( -h, 0), <Cm).

* This research was accomplished with the financial support of the RFBR (grant N 99-01-01079).

t Department of Mechanics and Mathematics, Moscow State University, Moscow, 117234, Russia

435

436 V.V. VLASOV

Let L:(.A) denote the matrix-valued function n

(3) L:(.A) = 2)Bi + .ADi) exp( -.Ahi); j=O

by l(.A) = det L:(.A), the characteristic quasi polynomial (see [1] for more details) of equation (1); by Aq, the zeroes of function l(.A) are numerated in increasing order of their modules counting multiplicities; by A, we denote the set of all zeroes of function l(.A).

The eigenvectors from canonical system of root vectors corresponding to characteristic number Aq are denoted by Xq,j,O (j = 1, 2, ... , r q), their adjoint vectors of orders- by Xq,j,s (s = 0, 1, ... ,pqj) (see [2] for more details).

We introduce the system of exponential solutions of equation (1):

(4) (t8 t•-1

) Yq,j,s(t) = exp(.Aqt) s!Xq,j,O + (s _ 1)!Xq,j,l + ... + Xq,j,s .

Denote by Wi,,((a,b),IC"') (-oo < a ~ (/ ~p(-2-yt) (t.ll"0' (t) Ill,.) dt) 'I', ., ;;, 0.

Here and everywhere in what follows W:\',0 = W:f, vUl(t) = :J:,v(t), p,j = 1, 2, ....

DEFINITION. A vector-valued function u(t) E W:f((-h,T),IC"') for arbitrary T > 0 is called a strong solution of problem (1), (2) if u(t) satisfies equation (1) almost everywhere on the semiaxis JR-r = (0, +oo) and condition (2).

LEMMA 1. Suppose that det D0 # 0. Then there exists "/'o ;) 0 such that for all"/';) "l'o problem (1), (2) is uniquely solvable in the space W:f,,(( -h, +oo ), IC"') for every vector function y(t) E WJ((-h,O),IC"'), and the following inequality

(5)

takes place with a constant d0 independent of function y(t).

Taking into account Lemma 1, we can introduce a semigroup U1 (t;) 0) of bounded operators acting in the space W:f(( -h, 0), IC"') according to the rule:

(6) (U1y)(s) = u(t + s), t;) 0, s E [-h, 0),

ON SPECTRAL PROBLEMS ARIZING IN THE THEORY OF FDE 437

where u(·) is the solution of problem (1), (2), corresponding to the initial function y(s) (see [3] for more details).

LEMMA 2. Let det Do # 0. Then the family of operators U1 (t ;? 0) forms a C0-semigroup in the space Wl((-h, 0), em) with generator llll, whose domain is

acting according to the rule: (lllltp)(s) = tpUl(s), s E (-·h,O).

PROPOSITION 1. Suppose that det Do # 0. Then the spectrum of the operator llll is the set A of the zer·oes of the junction l(A), and exponential solutions ( 4) are its root vectors and form a minimal system in the space Wl(( -h, 0), em).

LEMMA 3. Suppose that det Do # 0, det Dn # 0. Then there exist constants a 1 and a 2 such that the set A lies in the vertical strip {A : a 1 < ReA< az} and the system of exponential solutions {Yq,J,s(t)} is complete in the space Wl((-h,O),cm).

We denote by B(Aq, p) the disk of radius p > 0 with the center in the

point Aq, by G(A, p) = e \ ( U B(Aq, p)). >.,EA

LEMMA 4. Suppose that det Do # 0, det Dn # 0. Then there exists the system of contours fn ={A E e: ReA= az, Cn :( ImA :( Cn+!} U {A E

e: a1 :(ReA:( az, ImA = Cn+J} U {A E e: ReA= a 1, Cn :( ImA :( Cn+l} U {A E e : a 1 :( Re A :( a 2, Im A = Cn} belonging to the domain G(A, p) for wfficiently small p > 0. The following conditions are satisfied.

1. The values u+ = sup Re Aq, u_ = inf Re Aq, N = max vq are finite.

2. There exist constants 5, L:>. and a sequence of real numbers {en} : 0 < 5 :( Cn+!-Cn :( L:>. < +oo, n EN such that the system of segments rn = (t+icn), t E [u_ - /3, u+ + /3], /3 ;? 0 belong8 to the set G(A, p) for sufficiently small p > 0 and the number of roots Aq (counting their multiplicities) satisfying the inequalities

(7) u_ -- /3 < Re Aq < u+ + /3, Cn < Im A< Cn+l

is uniformly bounded by an absolute constant M.

438 V.V. VLASOV

3. There exists a constant Ko > 0 such that the following estimate is satisfied:

(8) sup 1-XII[.C-1(-X)II :( Ko, n E Z. >..Ern

THEOREM 1. Let det Do # 0, det Dn # 0. Then the system of subspaces Wn, where Wn is the span of all exponential solutions (4) corresponding to the numbers Aq satisfying inequalities (7), forms a Riesz basis of subspaces in the space Wi ( (-h, 0), rem).

In the next theorem we present the estimate of the strong solutions to problem (1), (2).

THEOREM 2. Let det Do # 0, det Dn # 0. Then for every strong solution u(t) of problem (1), (2) the following inequality holds:

llu(t + ·)llwJ(-h,o) = II(Uty)(s)llwJ(-h,O) :(

(9)

with a constant d independent on the initial junction y(t).

The following theorems make the previous one more precise in the case of a separate set A of roots {.Xq} of a quasipolynomiall(.X).

THEOREM 3. Let det D0 # 0, det Dn # 0. Assume also that the set A is separate, that is, inf 1-Xv- Aql > 0.

>.p#>., Then the system of sub spaces {V;.,}, where V;., is the span of all expo

nential solutions Yq,J,s(t) corresponding to the zero Aq, forms a Riesz basis in the space WJ((-h,O),rem).

COROLLARY. Under assumptions of theorem 3, the exponential solutions {Yq,J,s(t)} may be chosen in such a way that they form a Riesz basis of the space Wi (( -h, 0), rem).

THEOREM 4. Suppose that the conditions of theorem 3 are satisfied. Then for every strong solution u(t) of problem (1), (2) the following inequality is valid:

(10)

l[u(t+ ·)llwJ(-h,O) = II(Uty)(s)llwJ(-h,o) :(

:( d(t + 1)N-1 exp(xt)IIYIIwJ(-h,O)• t:;:, 0,

with a constant d independent of initial function y(t).

REMARK. It is well known that for quasipolynomials the constant N is finite. (See [4] for more details.)

2. Proof of the Main Results. We shall pay the main attention to the proof of theorem 1. Before that we formulate two propositions which will be used later.

PROPOSITION 2. Let det Do =f 0. Then the resolvent R(.\, ]!))) of the operator ]!)) may be represented in the following form:

(R(.\,Jl)J)j)(t) = -exp(.\t).C-1(.\) [texp(-.\hJ)DJf(O)-

t

(11) + exp(.\t) j exp( -.\r)f(r) dr, .$/A, t E [-h, OJ

0

and is a compact operator in the space W]( ( -h, 0), en).

PROPOSITION 3. Let det Do =f 0, det Dn =f 0. Then the matrix-valued function .c-1

(.$ satisfies the following estimates:

(12) II.C-1(.\)[[ ,; const([.\[ + 1t\ .\ E G(A, p) n {Re .\ > 0},

(13) II.C- 1(.\)[[ !( const([.\[ + 1t1 exp(Re.\h), .\ E G(A,p) n {Re.\ < 0}.

Proof of the theorem 1. Taking into account (11), we obtain the following representation:

- R(.\, D)f = exp(.\t)F(.\) + expyt) f(O)-

t

(14) -exp(.\t) J exp(-.\r)f(r)dr, f(t) E W]((-h,O),IC""),

0

where

440 V. V. VLASOV

In turn, we represent the vector-valued function F(.A) in the following way:

(16) F(.A) = -[Q(.A) + .c-1(.A)P(.A)],

where

n

P(.A) = L exp( -.Ahj)Gi(.A), j=l

0

(17) Gj(.A) =I exp(-.Ar)(Djf(ll(r)+Bjf(r))dr. -hj

Now let us estimate the vector-valued function F(.A). Note that the vector-valued functions Gj{.A) are entire functions of exponential type (not more than hj) and belong to Hardy space in every strip {A: A< Re.A < B}. Moreover, they satisfy the inequalities

+co

sup I IIGi(x + iy)ll2dy,; clllfii~J[-h,O]

A~x~B -co

with a constant c1 independent on function f(t).

(18)

Hence we obtain

+co

sup I IIP(x + iy)ll2dy,; c2IIJII~J[-h,o]

A;(x(B -co

with a constant c2 independent on function f ( t).

On the base of proposition 3 and trace theorem (see [7] for more details) we derive the following estimate of the function Q(.A) in the domain IIp(f3~> f3z) = G(A, p) n {A: ,131 < Re.A < f3z}:

(19) IIQ(.A)II ( c3(I.AI + l)-2 llfllw:}[-h,O]• c3 = const > 0,

where ,131, f3z are such constants that ,131 ( O<t, f3z ) O<z. Taking into account representation (16), proposition 3 and estimates

(18),(19) (for Re .A= D<p, p = 1, 2), we conclude that

+oo

(20) J (1 + lav + iMI 2)IIF(ap + ifl)ll 2df1 ( c411fll~:l(-h,o) -00

with a constant c4, independent on function f(t). Hereafter we need the following result from [5] (see also [6]).

LEMMA 6.5 [5]. If for every elements f and g E H = Wi(( -h, 0), en)

(21) I: J (R(.A, ID!)f, g)Hd.A < +oo, nEZ rh

then the subspaces Wh = Pn Wi ( ( -h, 0), en) <;; H = Wi((-h, 0), en), where Pn are the Riesz spectral projections of the operator ID! corresponding to contours r "' form an unconditional basis in the closure of its linear span. Moreover, if this system of subspaces is complete, then it is an unconditional (Riesz) basis in the space H.

Owing to Lemma 3, the system {Wn}nEZ is complete in the space Wi(( -h, 0), em). So let us verify inequality (21).

Due to ( 14), we have to prove that

(22) L j<exp(.At)F(.A),g(t))wi(-h,o)d>. < +oo. nEZ rh

Note that integrals of second and third terms in the right-hand side of formula (14) are equal to zero (except maybe one integral of the second term), because the integrands are holomorphic in the domains bounded by contours r h.

Hence we have

(exp(.At)F(.A),g(t))wJ(-h,o) = (.AF(.A),.ih(.A))cm + (F(.A),go(>.))cm,

442

where

V.V.VLASOV

0 0

YI(A) = J exp(At)g<1l(t) dt, -h

Yo(A) = J exp(At)g(t) dt. -h

In order to prove (22), it is sufficient to prove the inequalities

(23) L J (Ai F(A), YJ(A)) dA < +oo, j = 0, 1. nEZ rh

Vector-valued functions g1 (A), §o(A) are entire functions of exponential type (not more than h) and belong to Hardy space H2(A, B) in every strip {A : A :( ReA :( B}. Moreover, the following estimates are fulfilled:

+oo

(24) sup J ii9J(x + iy)ii 2dy :( KillgUliiL<-h,oJ• j = 0, 1, A(x(B

-00

~ with constants K0 , K1 independent on function g(t). So, for A = a 1 and B = a2 we obtain

"'•j•+'(Ai F(A), YJ(A)) dA ;;;

ap+icn

+oo :( J l((ap+iJ.L)iF(ap+iJ.l),gj(ap-iJ.l))l dJ.!:( -oo

with a constant c5 independent on function g(t). In turn, from inequalities (20), (25) we have

(26)

:L nEZ

"'•j•+'(AJ F(A), YJ(A)) dA

O:p+icn

with a constant c6 independent on functions f and g. Now we show that

a2+icn

(27) I: J (>J F(>.), fh(>.)) d), ~ C7llfllw~(-h,O) !lg(j)!IL,(-h,O) nEZ 0:1 +icn

with a constant C7 independent on f and g. In order to prove (27), we need the following proposition which is an

insignificant modification of theorem 3.3.1 from [7]. Denote by 9Ytv, (IR) the set of all entire functions of exponential type v,

which belong to the space L2 (R) as functions of real argument t E JR.

LEMMA 5. Let v(t) E 9Ytv2 (R), and let sequence of real numbers { tn}nEZ satisfy the condition: 0 < 6 ~ tn+l - tn ~ 6. < +oo, with certain positive constants 15 and 6..

Then the following inequality takes place:

(~ 11!(tn)JZ) 112

~ 15-1/2

(1 + v!l) (_£""1v(tWdt)

112

According to representation ( 14), we have

(>.JF(>.),9i(>.)) = -(>.JQ(>.),gJ(>.))- (,X.J.£:- 1(>.)P(>.),gJ(,\)).

Due to Lemma 4 and estimate (19),

(28) II-\Q(-\)IIJ1m-\=cn ~ Cs ~~~!;en (l-\1 + l)- 1 llfllw~(-h,O)·

From the last inequality we reduce the estimate a:z+icn

(29) j II>-Q(-\)II 2 Id>-l ~ c9(lnl + W2 llfll~~(-h,o)·

with a constant c9 independent on function f(t). In turn, according to Lemma 5 for the vector-valued function §1(,\), we

have +oo +oo

I: I(.Yz(x + ic.,), ejW ~ c10 j I(.Yz(x + iy), ejWdy ~ n=-oo -oo

+oo

~ c11 j ll9z(x+iy)!l 2 dy, x E [a1,a2], l = 0, 1, -00

444 V.V. VLASOV

where { ej }}=1 is an orthonormal basis of the space iC"'. Then, due to (24),

+oo a2 · +oo

n'fool 11.9t(x + icn)Wdx ~ C12 "'~~~"'I ii9l(X + iy)ii2dy ~

a1 -oo

(30)

with positive constants c12 and Ct3· Taking into account that functions (P(.A), ej) also satisfy the conditions

of Lemma 5, by analogy with estimate g1(.A), we obtain the inequality

+oo +oo

(31) L IIP(x+icn)llbn ~ C14 I IIP(x+iy)libndy, X E [a1,a2]. =-oo -oo

Then, according to estimates (32) and (18), we have

(32)

with a constant c15 independent on function f(t). From the last inequality and estimate (8) we obtain the inequality

+oo

(33) L I II.Ac-1

(.A)P(.A)II2

Id.AI ~ Ct611flliv~(-h,0)' n:::::::-oo ln

where ln ={.A E C: Im.A = Cn, a 1 ~ Re.A ~ a 2}. Taking into account representation (16) and estimates (30), (34), we

derive the inequality

(34) +oo

L I II.AF(.A)II2

Id.AI ~ Cl711JIIivi(-h,O)> n=-oo ln

with a constant c17 independent on function f(t). Hence from estimates (31), (35) follows the estimate

(35)


with a constant c18 independent on functions f and g.

At last, combining inequalities (26) and (36), we obtain that

(36) I: J (R().., JfJJ)j, g) d).. ( C!gllfllwJ(-h,O) llgllwJ(-h,O) n=-oo ln

with constant c19 independent on functions f and g.

So, according to Lemma 6.5 [5], the sequence of subspaces {Wn}nEZ forms an unconditional basis (Riesz basis) of the space W:}(( -h, 0), C").

Now we restrict ourselves to few commentaries about formulated results. Theorem 2 may be deduced from Theorem 1 of our paper and Theorem 1

of [13]. In turn, Theorem 3 may be obtained by strict analogy with Theorem 1. Lemmas 1 and 3 were proved in [8]. The proof of Lemma 4 is based on

certain results from [4] and [11]. Lemma 2 may be obtained by analogy with similar result from [10] (see also [3]).

3. Remarks and Comments. It is important to underline that estimates similar to (9), (10), where xis replaced by x+c: (c: > 0) are well-known (see [3], [10] for more details). Due to this reason, rather naturally the following problem arises. Is it possible to obtain more precise estimates for the solutions of equations of neutral type, and, in particular, to put c: equal to zero?

Theorems 2 and 4 give a positive answer (in a certain sense) to this question.

It is relevant to note that the asymptotic behavior of the solutions of equation (1) was studied by many authors. We restrict ourselves to drawing attention to monographs [1], [3] and paper [10].

We point out that our approach to studying this problem has a spectral character and is based on Riesz basisness of the system of exponential solutions of equation (1), which simultaneously forms the system of eigen- and associated functions of the generator JfJJ of the semigroup U, (t;, 0).

Similar results about Riesz basisness of the system of exponential solutions for differential-difference equations and the estimates of its solutions are presented in [8], [9], [14], [15]. Similar results for scalar equations of n-th order are proved in [12].

Under another understanding of solutions, basisness of the system of exponential solutions was investigated in the space £ 2 ( (-h, 0), C") Ell em in paper [16].

446 V.V. VLASOV

The completeness of the system of exponential solutions of equations similar to (1) was studied by many authors. We restrict ourselves to drawing attention to papers [17], [18] (see also references cited in these papers).

REFERENCES

[1] R. Bellman and K. Cooke, Theory of differential-difference equations, Academic Press, 1957.

[2] LC. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Nauka, Moscow, 1965 (in Russian).

[3] J. Hale, Theory of functional differential equations, Springer-Verlag, New York, 1984.

[4] A.M. Zverkin, Series of expansions of the solutions of differential-difference equations (Part I. Quasipolynomials) (1965). Proceedings of the seminar on the theory of differential equations with retarded argument (in Russian).

[5] A.S. Markus, Introduction into the spectral theory of polynomial operator pencils, Kishinev, 1968 (in Russian).

[6] N.K. Nikolskii, Treatise of the shift operator, Nauka, Moscow, 1980 (in Russian). [7] S.M. Nikolskii, The approximation of functions of many variables and embedding

theorems, Nauka, Moscow, 1977. [8] V.V. Vlasov, Correct solvability of a class of differential equations with deviating

argument in Hilbert space, Izvestiya Vuzov. Matematika, I (1996), 22-35. [9] V .V. Vlasov, Certain properties of the system of elementary solutions of differential

difference equations,Uspekhi Mat. Nauk, 51, 1 (1996), 143-144 (in Russian). [10] D. Henry, Linear autonomous neutral functional differential equations, J. Diff.

Equat., 15 (1974), 106-128. [11] B.Ya. Levin, On bases of exponential functions in L2, Zapiski Kharkov Mat. Ob

shch., 27, ser. 4 (1961), 39-48 (in Russian). [12] V.V. Vlasov, On a certain class of differential-difference equations of neutral type,

Izvestiya Vuzov. Matematika, 441, 2 (1999), 20-29 (in Russian). [13] A.L Miloslavskii, On stability of certain classes of evolutionary equations, Syb. Mat.

Zh., 26 (1985), 118-132 (in Russian). [14] V.V. Vlasov, On spectral problems arising in the theory of differential-difference

equations, Uspekhi Matem. Nauk, 53, 4 (1998), 217-218 (in Russian). [15] V.V. Vlasov, On properties of strong and exponential solutions of differential

difference equations of neutral type, Rus. A cad. Sci. Dokl. Math., 364, 5 (1999), 583-583 (in Russian).

[16] S.M. Verduyn Lunel and D.Yakubovich, A functional model approach to linear neutral functional differential equations, Journal of Integral Equations and Operator Theory, 27 (1997), 347-378.

[17] S.M. Verduyn Lunel, Series expansions and small solutions for Volterra equations of convolutions type, J. Differential Equations, 85 (1990), 17-53.

[18] S.M. Verduyn Lunel, The closure of the generalized eigenspace of a class of infinitesimal generators, Proc. Roy. Soc. Edinburg Sect., A 117A (1991), 171-192.

VOLUME 8 2001, NO 3-4 PP. 447-457

DIFFERENTIAL EQUATIONS ARISING IN QUEUING THEORY

N.D. VVEDENSKAYA AND YU. M. SUHOV •

We consider a boundary value problem for a system of differentialdifference equations. The problem is closely connected with some problems in queuing and network theory. Before presenting the equations let us say some words about the original network problem.

A popular class of network models is presented by Jackson networks. In such a model there exists a collection of J servers where tasks arrive and are served. After completing the service in server j, j = 1, ... , J, the task quits or remains in the network; in the latter case it joins a server i with probability PJ,i, i = 1, ... , J. We deal with generalization of such networks, namely with fast Jackson networks, where the dynamic routing is used. In the simplest case a network contains two stations S1, j = 1, 2, each station containing N single exponential servers of rate one. The model is determined by the exogenous Poisson flows >.1 and >. 12 of rates N >.1 and N >. 12 and routing probabilities

pj, Pi,i, PJ,if E [0, 1], Pj,l + PJ,2 + PJ,12 = 1, Pi+ p; < 2, ,j = 1, 2.

Upon arrival in flow Aj, a task randomly selects two servers in station S1;

upon arrival in flow, >.12 a task randomly selects one server in station S1

and another one in station S2 ; after that it chooses a server with the shortest queue and joins this queue. If there is more than one such server, the selection is performed at random. After being served by station j, the task quits with probability (1- pj) and remains in the system with probability pj. In the latter case it picks up a flow to S;. with probability PJ,i and the flow to both stations with probability p1,12 , then again selects at random two stations

• Institute of Problems of Information Transmission, 101447 Moscow, Russia

447

448 N.D. VVEDENSKAYA AND Yu.M. SUHOV

etc. As N -t oo, the model performance can be described by a solution of a boundary value problem for a system of differential-difference equations presented below (the limit N -t oo exploited in this paper is closely related to the so-called mean-field limit in statistical physics):

u;(l, t) = u;(l + 1, t)- u;(l, t)+

+(-A;+ L PkPk,iuk(1,t)) ((u;(l-1,t))2

- (u;(l,t))2)+

k=l,2

(1) + ( -A12+ L PkPk,l2uk(1, t)) (u;(l-1, t) -u;(l, t))(ui(l-1, t) +ui(l, t))/2, k=l,2

t > 0, l :::: 1, i = 1, 2, j of i,

(2) u;(O, t) = 1, t 2': 0, lim u;(n, t) = 0, t 2': 0, n-;oo

(3) u;(l, 0) = g;(l), l 2': 1.

In the network setting conditions

(4) 1 = g;(O) 2': g;(1) 2': ... 2': 0,

(5) 1 = u;(O) 2': u;(1) 2': ... 2': 0

have to be fulfilled. Therefore we usually presume that (4) take place. A particular attention is paid to fixed points of (1)-(3), i.e.,

2

a;(l)- a;(l + 1) = (.A;+ LPkPk,iak(1)) ( (a;(l- 1))2- (a;(l))2) + k=l

2

(6) + ( A12 + LPkPk,l2ak(l) )(a;(l- 1)- a;(l))(ai(l- 1) + ai(l))/2, k=l

l :::: 1, i = 1, 2, j of i,

DIFFERENTIAL EQUATIONS ARISING IN QUEUING THEORY 449

(7) ai(l) = 1, lim a,;(n) = 0. n-+oo

We present the conditions under which problem (6)-(7) has a solution and lim ·ui(l, t) = ai(l).

t-+oo

Solutions to (1)-(3) and (6), (7) are denoted by u(t) = (u1 (t), u2 (t)) and a= (a1 , a 2). We also denote

V(k, t) = L L Ui(l). i~l,2 l?:k

Let 11 be a space that consists of vectors g = (g1 , g2), gJ = (gJ(O), 9J(1), ... ), .9J(O) = 1, lim .9J(n) = 0, .9J(n) 2: .9J(n + 1) 2: 0. We endow 11 with the

n-+oo

distance p(g,g'), g = (g1 ,.92), g' = (g\,g'2 ), given by

(8)

this turns 11 into a complete compact metric space. The subset U c 11 consisting of those g E 11 for which

L.9i(l)<oo, i=1,2. l

In what follows the inequality g(ll 2: gC2l between g(ll, g(2) E 11 is understood entry-wise: g?) (l) 2: g~2\1), i = 1, 2. Given M E Z+, we denote by glMJ the truncated vector, with glM1(1) = .9i(1), 0 S: l S: M, 9i(M + 1) = 0.

THEOREM 1. a) For V g with 0 S: .9i(n) S: G < oo, problem (1) - (3) has a unique solution u(t), 0 S: ui(n, t) S: G, t ::0: 0, i = 1, 2.

b) If g E 11, then for V t ::0: 0 u(t) belongs to 11. c) If g E U, then for V t ::0: 0 u(t) belongs to U.

d) If g(l) 2 g(2), then u(ll ::0: u(2), where u(il are the corresponding solutions.

Tm;OREM 2. There exists at most one a E U solving (6), (7) and, if such a point exists, then for all g E U, lim u(t) = a.

t-+oo

Denote

450 N.D. VVEDENSKAYA ANDYu.M. SUHOV

and denote by l~s) the line

On a plane x = (xi> x2 ) consider a region:

R(s) : {0 <X! < 1, 0 < X2 < 1, ll9)(x) <X;, i = 1, 2,}

where

Denote

If s = 1, we omit index (s) to ease the notation.

CONDITION 1. A + pj + p~ < 2, and R is not empty.

THEOREM 3. If Condition 1 takes place and A12 = 0, then a E U and the solution to (1), (2) tends to a as t--+ oo.

If Condition 1 takes place, A12 > 0 and the line l0 intersects R, then a E U and a solution to (1), (2) tends as t -> oo to a, a;(1) E Rnlo, i = 1, 2. a;(n) decrease superexponentialy as n--+ oo, i = 1, 2.

Consider a truncated boundary value problem:

2

+ ( .\; + "L:>~Pk,iu~MJ (1, t)) ( ( u\MJ (l - 1, t))2 ( u\MJ(l, t) )2) +

k=l

2

(9) + ( .\12 + :~:::>kPk,!2U~M](1, t)) X

k=l

t > 0, 1 ::; l ::; M, i = 1, 2, j i i,

(10) u;(O, t)lMI = 1, u\M](M + 1, t) = 0, t 2: 0,

(ll)

The fixed points to (9)-(11) are

2

a)Ml(z)- a\Ml(z + 1) = (>.; + I.>ZPk,;a1Ml(1)) ((a\M1(l-1))2- (a\Ml(z)) 2)+

k=l

2

(12) + ( A12 + LPkPk,12a1M1(1)) (a\M] (l-1) -a\Ml (l)) (a1M1(l-1) +a 1M] (l)/2, k=l

1 ::; l ::; M, i = 1, 2, j of i,

(13) a;(1) = 1, a;(M + 1) = 0.

LEMMA 1. a) For V M E Z+ and for V g, with 0 S gi(n) S G < oo there exists auniquesolutionulMI(t,), t 2:0, to (9)-(11), andO::; u;(n,t)::; GVn<M+1, t>O,i=1,2.

b) Ifg obeys (4), then ulMI(t) obeys (5) Vt > 0.

c) If g(ll, gC2) obey gC1l 2: gC2l, the corresponding solutions ulMJ,(ll(t) and ulM],(2l(t) obey ulMJ,(ll(t) 2: ulMJ,(2l(t), Vt 2: 0.

Proof of Lemma 1. a) System (9) contains finitely many first order differential equations with smooth coefficients; the existence and uniqueness of the (global) solution to (9)-(11) as well as the continuity with respect to the initial data and to the system parameters follow from standard theorems. We have only to check that Vt u;(l) 2: 0 and max1 g;(l) 2: max1 u;(l, t), i = 1, 2, t > 0. (For simplicity, symbols [M] are temporarily omitted from the notation).

Let mint Uj(l, t) = Uj(lo, t), lo < M + 1, .i = 1, 2. Then u(lo, t) 2: 0, thus min1u1(l,t) 2:0. Let max1uj(l,t) = u1(l1 ,t), 11 > 0. Then ui(l1,t)::; 0, thus the value of max1 u1(l, t) cannot increase.

b) Owing to the continuity on g, we can assume until the end of the proof that g is such that g;(l) > g;(l + 1) < 1, 1 ::; l ::; M, i = 1, 2, and so are g(ll and gC2l, and furthermore, gpl(z) > gi(Z)(l), 1:; l::; M, i = 1, 2. We

will then check that with time the inequalities remain strict. Assume that t0 E (O,oo) is the first time when equality u;(l + 1,t) = u;(l,t) occurs for some i and 0 ::; l ::; M and pick up i0 , say io = 1, and 1° with this property. As u 1(0, t0

) = 1 > 0 = u1(M + 1, t0), we can always choose 1° so that either 1° ;=:: 1 and u1(l0 - 1, t0) > u1(l0 , t0) = u1(l0 + 1, t0), or 1° ::; M- 1 and u1(l0 , t0) = u1(l0 + 1, t0) > u1(l0 + 2, t0). In the first case it1(l0 , t0) > 0 and it1 (10 + 1, t0 ) ::; 0, which makes this case impossible. A similar argument is applied in the second case.

c) Now assume that t0 E (0, oo) is the first time when equality uP)(l, t) = ul2

) (l, t) occurs for some i and 1 ::; l ::; M and pick up i 0 , again say i 0 = 1, and 1° with this property. It is convenient to take the largest 1° for which the above equality holds; in this case we find, as before, that u\1

) (l0 , t0 ) > u\1) (l0 , t 0), which contradicts the choice of t 0• 0

LEMMA 2 For any gi, 0 ::; 9i(n) ::; G < oo, lim g;(n) = 0, i = 1, 2 n->oo

and for any t E [0, oo), there exists the limit in metric (8)

u(t) = lim u[Ml(t). M-too

Furthermore, u(t), t E [0, oo), gives a unique positive bounded solution to (1)- (3).

If g E a, then u(t) E a Vt. If g E U, then u(t) E U Vt.

Proof of Lemma 2. First observe that if u[Md(t) and u[M,J(t) are two solutions with initial data g(ll ;::: g<2l and u\M!](l0 , t) ;::: u\M'J(lo, t), i = 1, 2, for some lo ::; min [M1, M2] and V t ;::: 0, then u\Md(z, t) ;::: u\M'l(z, t) for V t ;::: 0 and 1 ::; l ::; l0 - 1. This follows when one compares the right-hand sides of corresponding systems (9) subsequently for l = l0 - 1, lo - 2, etc.

Next, note that u\M+l](M, t) ;::: u\MJ(M) , t ;::: 0 and use the above fact to get that u\M+l](l, t) ;::: u\Ml(z, t), t;::: 0, 1 ::; l ::; M. Continuing, we obtain the monotonicity in M and hence the convergence lim u\Ml(z, t) = ui(l, t).

A.f-too

The limit u satisfies (1)-(3). If g E a, then u E a. The uniqueness of the solution in a may be established by standard methods.

Finally, if g E U, we write the equations for V(l, t) =I.; I.; ·ui(T, t) i T?_l

(14) +h2(u(1, t))u1(l 1, t)u2(l- 1, t)- u1(l, t)- u2(l, t).

Here u(1, t) = (u1(1, t), u2(1, t)). As V t u(t) E 11, the right-hand side of (14) is uniformly bounded in time. Thus, V(1, t) grows at most linearly with t and therefore remains finite. This completes the proof of the Lemma. D

The proof of Theorem 1 follows from Lemmas 1 and 2.

LEMMA 3. a) For any M E Z+, there exists in U a unique solution alMJ to (12), (13) and V g E U, the solution ulMl(t) converges, a8 t--+ oo, to alMJ in metric (8).

b) If :J eo< 1 such that for V M > Mo a1M1(n) <eo for some j, n?: 1,

then lim a\M](1) = a;(1) < 1, i = 1, 2. M-+oo

c) If :J e1 < 1 such that for V M > Mo

l;(alMI(1)) < e1 , i = 1, 2,

/t(alMI(1)) + l2 (alMI(1)) + 112(alMI(1)) < 2

(here alMJ(1) = (a~J(l),a~J(1)) ), then there exist e, e < 1, SUCh that a\Ml(n) < en, and in this case a;(n) = lim a\MJ(n) decrease superexponen

M->oo tially as n --+ oo.

d) I:; a\M](1)(1- pi) :SA and alMJ increase in M. i=l,2

Proof of Lemma 3. a) Set gp(l) = 60~, i = 1, 2. Owing to the monotonicity of ulMI(t) with respect to the initial data, the entries u\Ml(z, t) = u\Ml(l, t) of a solution to (9)--(11) are nondecreasing in t, and as ulMI(t) obey (5), they are bounded from above by one. Therefore, there exist the limits a\MJ(l) =

tlim u\Ml(z, t), 1 :S l :S M, i = 1, 2, and limiting values a\Ml(l) obey (12), (13). -too

Next we show that a\Ml(z) = lim u\Ml(z, t) V g, 0 :S g;(n) :S G < oo; this t->oo

will imply the uniqueness of the fixed point alMJ. For simplicity, the index [M] will be temporarily omitted.

First, assume that g ?: a. Then u(t) ?: a V t. We will check that 00

J 2.:: (u;(l, t))- a;(l))dt < oo by using the induction in I, 0 :S l :S M. For 0 i=l,2 l = 0, we have this bound automatically owing to the boundary condition. Also, for l = 1,

V(1,t) =- L (t;(u(1,t))(u;(M,t)) 2 -l;(a(1))(a;(M)J 2)-

i=1,2


- L (u;(1, t)- a;(1)) :::; 0, i=l,2

i.e., V(1, t) remains bounded in t. Now assume the induction hypothesis:

V(l,t) = L (t;(u(1,t))(u;(l-1,t))2 -l;(a(1))(a;(l-1))2)+ i=1,2

- L (u;(l, t)- a;(l))-i=l,2

- L (zl(u(1,t))(u;(M,t))2- h(a(1))(a;(M)J2

)-

i=1,2

:::; ( L (z;(u(1,t))(u;(l-1,t)) 2 -l;(a(1))(a;(l-1))2)+ i=1,2

- L (u;(l,t)- a;(l)), i=1,2

t

V(l, t)- V(l, 0)):::; j( L (l;(u(1, s))(u;(l- 1, s)) 2- h(a(1))(a;(l- 1))2)+

0 t=1,2

(15) - "L)ui(l, s)- ai(l)J)ds. i=1!2

Owing to the induction hypothesis, the integral over terms with ui(l- 1, s) and ai(l- 1) is bounded. As the left-hand side of (15) is bounded uniformly in t, it follows that the integral

00

j 2)ui(l, s)- a,_(l))ds < oo.

0 z=l 12

A similar integral converges when g ~ a. Thus, if either g ~ a or g 2: a, the solution u(t) approaches a as t-+ oo. In the general case we pass tog+ = max [g, a] and g- = min [g, a] and use the monotonicity of u(t) in g. This completes the proof of convergence u(t) -+a. The convergence implies the uniqueness of a.

b) Owing to (12), a\M1(1) < 1, because otherwise 'r!n ai(n) = 1. Further, a,(n)-a,_(n+1) ~ C(ai(n-1)-ai(n)) where C < oo, and if lim a\Ml(1) = 1,

M-too

then for any n, E there exists a number Mn such that a\M](n) > 1 - E,

M > Mn· That contradicts the condition a\Ml(n) < 00 .

c) Summing equations (12), (13) for l = n, ... , M, we get (the index [M] is again temporarity omitted)

a1(n) + az(n) = h(a(1))ai(n -1)+

For 0 ~ x, y ~ 1 and c1 , c12 , c2 > 0, the inequality holds:

Thus by (16) we have

(17)

() = max[h(a[Ml(1)), 12(a[Ml(1)), (l1(a[Ml(1)) + h(a[Ml(1)) + h 2(a[Ml(1))/2]. It follows from (7) and (17) that for V iJ,() < iJ < 1, a;(l) = lim a\Ml(l) <

M-+oo -2! CO , C =const.

d) The bound for I: a\MJ(1)(1- pi), is derived from (12), and alMl(n) i==1,2

increase in M because u!Ml(n), n:;:: 1, increase in M. 0

The proof of Theorem 2 follows the proof of Lemma 3.

LEMMA 4. Let a E U. If A12 = 0, then a;(1) = l;(a(1)). If A12 > 0, then a;(1) E 10 n R.

Proof of Lemma 4. Summing (6) for l = 1, 2, ... , we get

a1(1)(1- pr) + a2(1)(1- p;) =A,

00

a;(1) = l;(a(1)) + L l12(a;(l- 1)- a;(l))(aJ(l- 1) + aJ(l))/2. 1=1

This proves the Lemma. 0

Return to s, 0 ::; s ::; 1 and consider systems (1) and (6), where spj stands for pj, i = 1,2, 0::; s::; 1. Denote by u<'l(t), a(s) the corresponding solutions.

PROPOSITION 1. Let Condition 1 be fulfilled. If the intersection £(1) :

R((1) n 1~1 ) is not empty, so are the intersections L(s) : R(s) n 1~'), 0 ::; s ::; 1.

The values of l)'l(x), x E £(s), are bounded by()*< 1, uniformly ins.

The proof is omitted.

Proof of Theorem 3. We have only to prove that a E U. For A12 = 0, the proof is straightforward: really, a;(1) = li(a(1)) < 1 i = 1,2, and therefore a1(n) + a2(n) < B(a1(n- 1) + a2(n- 1)), () < 1. Let A12 > 0 and let Sk = ko, k = 0, 1, ... 1/o, where o > 0 is sufficiently small.

First show that a)'l (l) increase in s, i = 1, 2, l 2': 1. Consider u'1 with initial conditions u'1 (0) =a'', s1 > s2 (a<sz) E U.) Then itf1 (l, 0) > 0, thus u'1 increases in t and therefore a)'1)(l) > a)''l(z), l > 0.

Owing to the equality a\')(1)(1- spi) + a~')(1)(1- sp2) =A (a;(n) = lim a\Ml(n)) a)'l(1) are uniformly continuous ins.

M-+oo

Use the induction in k. As l)'l do not depend on a;(1) as k = 0, s = 0, V M a\MJ,(O) ( l) ::; ()1, i = 1, 2. Using Lemmas 3c and 4, we prove the Theorem for this case.

Now assume the induction hypothesis. If the Theorem holds for k ::; k0 ,

then a<••ol E L<••ol z(••ol < 0* and for /j sufficiently smallt<••o+d (afMl(l) < 1 ~ ' t t '

M--+ oo) z;••o+') < 1 because of uniform continuity of a)•)(l) and l;(a<•l(l))

in s. Thus, using Lemma 3bc, we get that a<••o+~l E U and therefore ai'•o+d E

L<••o+~l, z;••o+~l < 0*. That completes the proof. :D

REFERENCES

(1] N.D. Vvedenskaya, R.L Dobrushin and F.I. Karpelevich, "A queuing system with selection of the shortest of two queues: an asymptotical approach", Problems of Information Transmission, 32 (1996), 15-27.

(2] M.Mitzenmacher. "The J'\"""er of Two Choices in Randomized Load Balancing", PhD Thesis, Uni.,_ity of California, Berkeley, 1996.

[3J N.D. Vvedenskaya and Yu.M. Suhov, "Dobrushin's mean-field approximation for a queue with dynamic routing", Markov Proc. Rel. Fields, 3 (1997), 493-527.

(4] J.B. Martin, Yu.M. Suhov, "Fast Jackson networks", To appear in Ann. Appl. Prob., 1999.


VOLUME 8 2001, NO 3-4 PP. 459-472

HOMOGENIZATION OF PROBLEMS WITH FRICTION FOR ELASTIC BODIES WITH RUGGED BOUNDARIES *

G. A. YOSIFIAN t

Abstract. The system of linear elasticity is considered in a domain whose boundaw depends on a small parameter e > 0 and has a rugged part, which may bend sharply anil embrace cavities or channels, and as e -+ 0, it approaches a limit surface on the boundal;\y of the limit domain. Nonlinear boundary conditions characterizing friction are imposed on the rugged part of the boundary. We investigate the asymptotic behavior of solutions to such boundary value problems for variational inequalities as e -+ 0 and construct the limit problem according to the form of the friction forces and their dependence on the parameter e. In some cases, this dependence results in additional restrictions on the s.et of admissible displacements for the limit problem, which has the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone is described iin terms of the functions involved in the nonlinear boundary conditions on the rugged pant of the boundary and may depend on its geometry, as shown by examples.

Consider a sequence of bounded domains ne depending on a small parameter E > 0 and "approaching from outside" a fixed domain no, i.e., ne :) no, ne \no C {x E JRn: dist (x,an°) :0::: b(c:)}, 8(c:)-+ 0 as E-+ 0.

For an elastic body occupying the domain ne, the stress tensor at a poiJUit x = (x1 , ... , Xn) is a (n x n) matrix u(u') = A(x)e(u'). Here u' = u'(x)) is the displacement identified with the column vector '(ui, ... , u~), e(u') is the linearized strain tensor, i.e., the matrix with the elements (e(u'));j = 2-1

( auflaxj + aujjax; ); A(x) is the elasticity tensor identified with a linernr transformation of the space M" of real (n x n) matrices, i.e.,

' This work has been supported by the INTAS Grant No 96-lOin and Grant No 98-01-00450 of the Russian Foundation for Basic Research of the Russian Academy of Sciences.

t Institute for Problems in Mechanics, Russian Academy of Sciences, Prospekt Vernadskogo 101, Moscow, Russia, 117526

459

460 G_. A. YOSIFIAN

where the coefficients atk ( x) are bounded measurable functions in JRn satisfying the usual conditions of symmetry and positive definiteness:

hk( ) kh( ) ik ( ) a;j x = aji x = ahj x ,

for all real symmetric matrices {b;j} E sn and all x, where ~1 , ~2 = const > 0. Inside n•, the displacements are supposed to satisfy the usual equa

tions of static equilibrium div o-(u') = -F(x) with external body forces F E (L2(n))n. On a set r:D c an· n ano that does not depend on £ and

has a nonzero surface measure, the body is assumed clamped: u'IE = 0, D

and there is a set r:~ c an• \ 8n° on which we impose nonlinear bound-ary conditions, such as those expressing the Coulomb law of contact with friction, or the conditions of normal displacement with friction (see [1] and Examples 1 and 2 below); on the rest of the boundary, the body is free of traction. In accordance with [1], we formulate such boundary value problems in terms of variational inequalities involving convex nondifferentiable functionals; in other words, we consider weak solutions of these problems. Formal (i.e., regardless of regularity) equivalence of weak and "classical" solutions of problems with friction is discussed in [1]. Existence and uniqueness of weak solutions to variational inequalities considered in this paper follow from general results to be found, for instance, in [2] (see also [1]).

The present paper continues the studies of [3], [4], where some other homogenization problems with nonlinear boundary conditions have been considered, and where additional references can be found.

We are going to study the asymptotic behavior (as£ -+ 0) of solutions of the following problem for a variational inequality:

Find u• E HJ(ne, r:0

)n such that for any v E HJ(n•, r:0 )n

e e

(1) J e(v-u'):A(x)e(u')dx+J'(v)-J'(u') 2:: J F·(v-u')dx. o n

Here HJ(n',r:0 ) = {u E H1(n•) : u = 0 a.e. on r:0 }; H 1(n') is

the Sobolev space with the norm llui!I.o• = !lull£2(0') + !IV., ull£2(0'); J'(v) are convex continuous (nondifferentiable, in general) functionals on H 1(n•)n , specifying boundary conditions on r:~ and defined below; p: q = Pijqij for

p, q E M", and ' ·11 = (;7J; for ', 11 E lR". In order to pass to the limit in (1) and formulate the limit problem, we

make the following assumptions on the domains n• and the functionals J'.

HOMOGENIZATION OF PROBLEMS WITH FRICTION 461

From now on, we restrict ourselves to the domains n• with a rugged boundary near a plane, i.e., the set 80.' \ on° is supposed to consist of finitely many Lipschitz surfaces lying in a neighbourhood of a ( n - 1 )-dimensional planar domain f 0 . More precisely, consider a bounded Lipschitz domain 0.' C JRn = {x = (x, Xn)} that depends on a small parameter e > 0 and consists of two parts:

n'=int(0.0 UG'), 0.0 c{xn<O}, G'c{O<.Tn<e},

where no and G', are Lipschitz domains adjacent to a planar Lipschitz domain f 0 c 80.0 n {xn = 0} independent on e, and G' has the following structure. Let

l,={z=(z,O)E'lln: c(D+z)n{xn=O}cr0}, Q'=int Ue(D+z),

zElf;

where zn consists of z = (z1 , ••• , Zn) with integer Zj; 0 =]0, l[n; A+ z is the shift of a set A by a vector z E zn; eA = {x: c 1x E A}. It is assumed that:

(i) G' c Q'; G' n e(D + z) ,P 0, Vz E I,; oQ' n {xn = 0} coG';

(ii) G' is a Lipschitz domain formed bye-shifts of finitely many Lipschitz domains eR1 , ... , eRN, where Rj c D and N do not depend on e. By definition [3], this means that for every z E J., there exist j E {1, ... , N} and a E zn such that the cell G' n e(z +D) coincides with e(Rj +a).

The set r• = oG' \ { Xn = 0} which lies in the layer { 0 < Xn :; e} is called the rugged part of the boundary of the domain 0.'. The set oQ' n { Xn = 0} c f 0 coincides with oG' n {xn = 0} by assumption, and is denoted by r~.

For each cell a; = G' n e(D + z), consider the subsets of its boundary

Clearly,~~·· is a unit (n- I)-dimensional cube in the plane {xn = 0}, while ,;,e is an element of the rugged part of the boundary that consists of finitely many Lipschitz surfaces, which may embrace cavities or channels. Thus,

U "'O,e = fO tz e '

zElf;

Boundary conditions of friction type on the rugged part r• c an• will be described in terms of the following classes of functions (cf. [3]):

462 G. A. YOSIFIAN

Class Ap(lRn, 1, 3) consists ofreal-valued functions a(71, y, ~) defined on JRn x 1 x 3, measurable in~ E 3 for any 11 E lR", y E 1, and such that

supja(71,y\~) -a('q,y2,~)j :S: ~t(l + I11D jy1 -y2 j, Vy1,y2 E 1, 71E JRn, e

sup ja(711, y, ~) - a(172

, y, ~)j :S: It !111 - 712 j , V 11\772 E lRn .

v.e Class O(f") consists of sequences of functions g" ( 71, x) on JRn x r• such

that for almost all x E r• = UzEJ. ')';'• and all 11 E JRn, the restrictions of g"(71,x) on the rugged part of the boundary have the form

(2)

g~(71,fj,~) E A~'(JR;;,f'~'",c- 1 ')'!'"),

with ft = const > 0 independent of c, z; and there is a function 9(71, x) on JRn x f 0 , independent on c and such that

(3) J 9~(71, x, ~) dSe = 9(71, x) , (x, O) E /'~'" (a. e.),

e-1'/';'e

for all sufficiently small c > 0 and all z E I,. The function 9(71, x) is called the weighted mean value with respect to the fast variable for the sequence g" ( 71, X). (Note that the set of X E f" for which (2) holds does not depend on 71.)

On the space H 1(0")n, consider the following functionals:

jg(v) = J 'l10(v,x)dS, H(v) = J 'l!Hv, x) dS,

r' r'

where 'llij(71, x) and '111(71, x) are sequences of class O(f"), with 'llo(71, fj) and '11 1 ( 71, fj) being their weighted mean values in the fast variable, and 'l!Q.(71, fj, .;) and 'l!L(71, fj, ~) being the representations (2) of their restric-

' ' tions to the subsets ')';·•. It is assumed that both

(4) 'llij,.(71, fj, .;) and '11!,.(71, fj, .;) are convex with respect to 71,

and 'llij,z satisfies the additional conditions:

(5) 1jf5,z(11,Y,~) = 0 a.e. in ~ E C1/~'5 =? 1ji5,A77,!1,0 = 0, V ~ E c:-111'".

Moreover, we assume that the dependence of the closed convex cone

Mx={7JElRn: 1jio(7J,x)=O}

on x E f 0 is Lipschitz continuous in the sense that

where 'll'(7J; M) is the projection of rp E IRn to a closed convex cone M, i.e., 'IT(rp; M) is the unique vector in M on which inf(EM IC- 111 is reached. In particular, (6) holds if 1ji0 does not depend on x.

As the functional J 5 ( v) in (1), we take

(7) J'(v) = f..l.o(c:)jg(v) + f..I.I(c:)jf(v),

where f..l.o(c:) and 111(c) are real parameters such that

(8) 0 < f..l.o(c:) -too, 0 :<:; f..1.1(c:) -t a< oo as c -t 0.

The main result of this paper is that under the above assumptions the limit problem for (1) has the form of a variational inequality over the domain S1° with the functional J' ( v) replaced by the "limit" of the functional f..1. 1(s)jf(v), whereas the limit effect of the term f..l.o(c:)jg(v) amounts to imposing additional restrictions on the set of admissible displacements.

THEOREM 1. Let u' be the sequence of .solutions of problems (1), (7), (8). Then llu'- u0 ll 1,no -t 0 as c -t 0, where u 0 E V 0 is the solution of the variational inequality

(9) j e(v- u 0): A(x)e(u0

) dx + ](v)- ](u0) 2': j F · (v- u 0

) dx,

~ ~

for any v 0 E Vo = {v E HJ(S1°,E0 )n: 1j!0 (v(x,O),x) = 0 a.e.on f 0 },

where the set of admissible displacements V 0 is a closed convex cone in H 1 (S1°)n and ]( v) is a convex continuous functional on (H 1 (S1°) )n of the form

](v) =a j 1j! 1(v(x,O),x)dx.

ro

464 G. A. YOSIFIAN

· Before proving this theorem, we formulate some auxiliary results, which are special cases of more general facts established in [3] about functions from Sobolev spaces over partially perforated domains.

The following lemma holds in consequence of Lemma 5.2 in [3] and allows us to compare traces on the "oscillating rugged" set r• and traces on the planar domain r 0 for nonlinear functions of the elements of H 1(fY)n. It should be observed that although r• is close to r0 ' the classical theorem about the closeness of traces cannot be applied, because the set r• may have sharp bends, embrace cavities or channels, and cannot, in general, be suitably transformed into the planar domain r 0• In what follows, by C, Ci, etc., we denote constants independent on c:.

LEMMA 1. Let g"('q,x) be a sequence of class O(r•) with the weighted mean value g( 11, x). Then the following uniform estimate holds for any v E HI(Q•)n :

I J g"(v, x) dS- J g(v(x, 0), x) dxl s; C c:~ (llvllt,n• + 1) . re ro

The following two lemmas about extension operators and Korn's inequalities are consequences of Theorem 3.9 and Lemma 3.11 from [3].

LEMMA 2. There are linear extension operators pe : H 1(Q•)n -7 H 1(0)n from the domainn• toO= int(Q0u(r0 x[O,l])), such that sup. IIP"II < oo.

LEMMA 3. The Kom inequality holds uniformly inc::

llull 1,n, S:: Clle(u)llo,n•, VuE HJ(n•, Eot.

Our next statement is an analogue of Lemma 5.5 from [3] and is proved in a similar way on the basis of the assumed properties of the functions W"0(11, x).

LEMMA 4. Let W(i("l, x) be a sequence of class O(r•) satisfying conditions (4), (5), (6). Then

Vo = { v E HJ(0°, E0 t: Wo(v(x, 0), x) = 0 a.e. on r 0} =

- { v E HJ(n°,E 0 )n: j W"o(v(x,O),x)d!i: = 0}

:-·-~·· '.,


is a closed convex cone in H 1(f!0 )n. Moreover, any v 0 E V 0 admits an extension Y 0 E H 1(fi)n to the domain fi = int ( fJ0 U (f0 X [0, 1])) such that Y 0 does not depend on E and 1]!0 (Y0

( x), x) = 0 a. e. on re for E small enough.

Proof of Theorem 1. Taking v = 0 in (1), we obtain the inequality

ah(u',u') + J.Lo(E)jtJ(u') + J.LI(E).ii(u')- J.LI(E)ji(O) S (F,u')L'(fl'),

• where ah(u, v) =I e(u): A(x)e(v) dx. Hence, using the Korn inequality

n and the properties of IJ!i(17, x), we obtain the uniform estimate

(10) llu'll~,n" + J.lo(E)j0(u') S Co,

since Lemma 1 yields

1/ IJ!f(u',x)dS- J IJ! 1(u',i:)dxl ::;Ce~(llu'lh,nd-1), re ro

and therefore, l.if(u')l::; C1 (1 + llu% 11,) by the trace theorem on f 0• ,

Thus, for the extensions from Lemma 2, the norms IIP'u'll 1,n are boun-

ded uniformly in E, which implies that there is u 0 E HJ(n, E0 )n such that for a subsequence of E-+ 0 (still denoted by E), we have

where -' and -+ denote weak and strong convergence, respectively. Dividing (10) by J.lo(c) and passing to the limit as E'-+ 0, we find, using

(8) and Lemma 1, that J i!fo(u0 , i:) di: = 0, and therefore, u0 E V 0 by ro

Lemma 4. Let v 0 be an arbitrary element of Vo with its extension Y 0 E (H1(fi))n

from Lemma 4. Note that

- a 11o(u0, u 0) 2 lim sup (- an(u', u')), j 0(Y0

) = 0, j0(u') 2 0, £-tO

lim J.LI(c)Ji(u') = i(u0), lim J11(e)ji(Y0

) = ](V0),

e-tO e-+0

where the first inequality follows from the lower semicontinuity of the quadratic form ano(w, w) with respect to weak convergence in H 1(fJ0)n; the last

466 G. A. YOSIFIAN

two relations follow from Lemma 1. Now, using (1) with v = Y0 J~, we find

ano(u0, v0 - u 0) + J(v0)- J(u0

) =

=- ano(u0, u0) + ano(u0, v0) +lim Jll(c) (j~(V0)- jf(u')) 2: ..... o

2: lim sup (- ah(u',u') + ano(u', Y 0)) +

..... o

+lim sup (J!1(c:) (j~(Y0)- jf(u')) + J!o(c:) (j0(V0)- j 0(u'))) 2:

..... o

2: lim sup (antu•, Y 0- u') + J'(Y0

)- J'(u')) 2: ..... o

2: lim sup (F, Y0 - u') L'(!l') = (F, v0

- u0) L'(!JO) ,

..... o

which means that u0 is a solution of problem (9). Let us extend u 0 as U 0 E H 1(fi)n so that j 0(U0

) = 0 (see Lemma 4). Since j 0(u') 2: 0, from (1) with v = U0 J~ we get

0:::; aO(u'- U 0,u'- U 0) =

= an(u',u'- U 0)- a~(U0,u'- U 0

) =

= -( an(u',U0 -u')+J'(U0)- J'(u'))- a~(U0,u'- U 0

) + + f.to(c:)j0(U0

) + f.t1(c:)jf(U0)- f.to(c:)j0(u')- f.ll(c:)ji(u'):::;

:::; -(F, U 0- u')L'(W)- a~(U0 , u•- U 0

) + + J,ll(c:)jf(UO)- f.tl(c:)j~(u').

The right-hand side of the last inequality tends to zero as c: -t 0, and therefore, by the Korn inequality, llu'- u0 ll 1 no -t 0.

Since the solution of problem (9) is 'unique, the above reasoning can be applied to any subsequence of u•, and we come to the same solution u 0 • D

REMARK 1. Theorem 1 admits generalizations. For instance, other boundary conditions, like those of Signorini type (see [3]), can be considered on an• \ r•. In that case additional solvability conditions have to be imposed and taken into account while passing to the limit as c: -t 0. For the sake of simplicity, we have restricted ourselves to zero Neumann conditions on 80! \ (:1:0 ur•) and zero Dirichlet conditions on :1:0 , which ensure coerciveness of the corresponding quadratic form.

Another generalization of Theorem 1 is obviously obtained if instead of the functional J'(v) of the form (7) we take

K

J'(v) = L (f.J,o,a(c:)jij,,(v) + f.tl,a(c:)jf,,(v)) , a=l

HOMOGENIZATION OF PROBLEYIS WITH FRICTION 467

where 0 < Jlo,a(c:)-+ oo, 0::; Jll,a(c:)-+ aa < oo as e-+ 0, and

jg,a( V) = l iJJ6,,( V, x) dS, jf.a( V) = l iJJ1,a( V, X) dS ,

with Wo,a and iJ!i,a of the same type as iJ!0 and iJJi considered above. Assume that the dependence of the cone

Mx = { '17 ERn : iJ!o,a('l7, x) = 0, a:= 1, ... , K}

on x is Lipschitz continuous in the sense of (6). Then the limit problem for (1) is the variational inequality (9) with the set of admissible displacements Vo and the functional J( v) of the form

Vo={vEHJ(0°,I:0 )n: Wo,a(v,x)lro=O, a:=1, ... ,K}, K

J(v) = ~::>"' / iJ!l,a(v(x,O),x)dx. a=l ro

Examples of Problems with Friction. For normal and tangential components of displacements and stresses on 80', we use the following (formal) notation:

CJN(u) = v' · (cr(u)v'), crT(u) = cr(u)v'- CJN(u)v',

UN= U "Ve, UT = U- UNV£,

where v' is the unit outward normal to 80'. Suppose that the elastic body occupying 0' may have contact with ob

stacles along a given set E~ont on the rugged part of the boundary f', and

(11) ~~ont = U 8~' s' c 'Vl,e Z IZ l

zEI.,

where s; are nonempty sets, open in the topology off' = U 'Y;''. zEle

(12)

EXAMPLE 1. Consider the following problem:

- div (A(x)e(u')) = F in 0',

u' = 0 on E0 , cr( u')v' = 0 on 80' \ (E0 U E~ontl ,

CJN(u') = Jll(c:)cp' on E~ont,

for x E E~ont the following implications hold:

lcrT(u')l < Jl(c:)'I/J' =* u~ = 0,

lcrT(u')l = ft(c:)'I/J' =* :3.\2 0: u~ = -.\crT(u').

468 G. A. YOSIFIAN

Here Jtt(e) and Jt(e) are nonnegative parameters; the functions <P'(x) E L00(r') characterize normal stresses; 0:;; '1/J'(x) E L""(r•) describe friction forces. A mechanical interpretation of such problems in terms of two-sided contact with friction on ~~ont governed by the Coulomb law is given in (1], together with the justification and the definition of weak solutions considered here.

Let us introduce the following continuous convex functional on H 1 (fl')n:

J'(v) = Jt(e) j '1/J'(x)lvTidS ~ Jtt(e) j <P'(x)vNdS =

:E~ont :E~ont

(13) =Jt(e) j 'I/J'(x)lv~(v·v')v'ldS~Jt1 (e) j <P'(x)v·v'dS.

~~ont :E~ont

By definition, a weak solution of problem (12) is u' E HJ(fl', ~0 )n satisfying the variational inequality (1) with the functional (13).

In order to apply Theorem 1, let us consider more closely the functional (13) and make some additional assumptions. Suppose that

1/J' = <P' = 0 on r• \ ~~ont ' <P'(x) = ¢;(x) on s; ' 1/J'(x) = 1/;;(x) on s; , 1/;; 2: x:0 = const > 0, z E /, .

Then, (13) becomes

(14) J'(v)=Jt(e) j 8'(v,x)dS+Jt1(e) j A'(v,x)dS

where

re re

{ 1/J;(x) 111 ~ (71·v'(x))v'(x)l ,

8'(17,x) -0, xEr'\~~ont>

{ -<P~(x)(71· v'(x)) , xEs~ ,

i\'(71,X) = 0, xEr'\~~ont·

Suppose that the integrals

XEs~,

10n1_1 J 1/;;(x) 111- (71• v'(x))v'(x)l dS and

10n1_1 J ¢;(x)(71• v'(x)) dS

s• z s< '

HOMOGENIZATION OF PROBLE\1S WITH FRICTION 469

depend neither on z nor c. (In particular, this condition holds if the set ~~ont has an c-periodic structure, i.e., s~ = c(s + z), z E I., for some surface s, whereas the functions '1/J£ and ql on ~~ont coincide with c-periodic functions of the form 'I/J(c-1x), <jJ(c1x), respectively.) Then the sequences 8£(11, x), 11.'(11, x) have weighted mean values with respect to the fast variable, viz.,

A(17) cL j <P~(x)("l} • v£(x)) dS. s§

Our aim is to describe the limit behavior of the weak solutions u£ for

0 :S M(c)--+ b, 0 :S 111 (c)--+ a< oo as c--+ 0,

and illustrate its dependence on the geometry of ~~ont. Consider two qualitatively different cases: b < oo and b = oo.

1) If b < oo, then by Theorem 1 and Remark 1, the limit problem for (1), (14) is the variational inequality (9) with

Vo = Hfj(0.0, ~0 )n, J(v) = J (b8(v) + aA(v)) di:.

ro

2) If b = oo, then the limit problem for (1), (14) is (9) with

Vo = { v E Hfj(0.0, ~0 )n: 8(v)[ro = 0} , J(v) =a J A(v) di:.

ro

Consider more closely the set of admissible displacements Vo in the care of n = 3. Since '1/J~(x)?: Ko > 0, the relation 8(17) = 0 means that

1J = (11·v£(x))v£(x) a.e. in xEs~.

If s; contains three planar pieces on which v" takes three linearly independelllit values a 1 , a 2 , a 3 , then the implication 8(17) = 0 =? 11 = 0 holds, aru!l therefore on f 0 we obtain the Dirichlet condition v[ro = 0, which mea!il's that in the limit the body subject to friction becomes clamped along f 0 .

If s; consists of two planar regions with linearly independent normals a 1

and a 2 , then in the limit we obtain the boundary conditions

v[ro · a 1 = v[ro · a 2 = 0.

470 G. A. YOSIFIAN

If s; consists of parallel planar regions on which the normal v• can take only two values a 1 and -a1

, then

'TJ = (TJ·a1)a1 -<==? TJ•a2 = TJ•a3 = 0,

where a 1, a 2, a 3 form an orthonormal basis in JR.3 . In this case, the boundary condition on r 0 becomes

I 2_ I a_o Vro•a -Vro•a- ,

i.e., the tangential components of admissible displacements must vanish.

ExAMPLE 2. Consider a problem that describes normal displacement with friction (see [1]) on I:~ont , namely,

- div (A.(x)e(u•)) = F in n• , Ue = 0 On I;D , IT(U<)v< = 0 On ane \ (I:o U I;~ont) ,

ITT(u•) = 0 on I:~ont,

(15) for x E I:~nt the following implications hold:

-tLJ(e)gf(x) < aN(u•) < f.L2(e)gHx) => u~ = 0,

aN(u•) = -tLJ(e)gi(x)

aN(u•) = f.L2(e}gi(x)

=> u~ 2': 0,

=> u• <0. N-

Here /-Lj(e) 2': 0 are real parameters, gj 2': 0 (j = 1, 2) are functions in L00 (f<) characterizing friction forces. To define weak solutions of this problem, we follow [1]. On H 1(n•)n, consider the convex continuous functional

J•(v) = J (f.LI(e)gi(x)v~ + f.L2(e)g~(x)v~) dS = E~nt

(16) = J (f.LI(e)gi(x)(v ·v•)+ + f.L2(e)gHx)(v ·v•)-) dS,

E~ont

where a+= max{a,O}, a-= max{-a,O}.

By definition, a weak solution of problem (15) is u• E HJ(O•, I:0 )n satisfying variational inequality (1) with functional (16).

Consider sets (11), and let

gtzCx) = gi(x)l , ·~

g~,z(x) = gHx)L . •

Then functional (16) takes the form

(17) J'(v) = fti(c) j GHv, x) dS + ft2(c) j G~(v, x) dS, re p

where

Suppose that the integrals

£;_1 J 9Lz(x)('IJ · v"(x)j+ dS, f:L / g~,z(x)('IJ • v"(x))- dS

~ ~

depend neither on z nor c: (which is the case if the set L:~ont has an c:-periodic structure, i.e., s~ = c:(s + z), z E I., for some surface s, whereas gi(x) and gHx) on L:~ont coincide with t:-periodic functions). Then

s' '

G2('17) - c:L j g~,z(x)(Tf · v"(x))- dS

s:

are weighted mean values for the sequences Gi(Tf,x), G~(Tf,x). Suppose that

and consider three qualitatively different cases: 1) a 1 < oo, a2 < oo; 2) a 1 < oo, a2 = oo; 3) a 1 = a2 = oo.

1) If a1 < oo, a2 < oo, then by Theorem 1 and Remark 1, the limit problem for (15) (i.e., for (1), (17)) is variational inequality (9) with

Vo = HJ(0°,L: 0 )0

, f(v) = f (a1G1(v) + a2G2(v))di.

ro

472 G. A. YOSIFIAN

2) If a1 < oo, az = oo, then the limit problem is (15) with

J(v) = a1 J G1(v)dx. ro

3) If a1 = az = oo, then the limit problem is (15) with

J(v) =: 0, Vo = {v E HJ(0°,~0)n: G1(v)lro = Gz(v)lro = 0}

Note that the limit boundary conditions, such as G2 (v)lro = 0, depend on the geometry of the set ~~ont· Indeed, suppose that on s; the normal v" is continuous and gL > 0. Then the condition G2 (71) = 0 implies that "1' v"(x) :2': 0 for all ~ E s;. Therefore it is easy to construct the sets s~ in such a way that G2 (v)lro = 0 amounts either to zero Dirichlet condition for displacements in some directions, or to a condition of Signorini type ( cf. Example 1).

REFERENCES

[1] G.Duvaut and J.-L.Lions, Les Inequations en Mecanique et en Physique. Dunod, Paris, 1972.

[2] I.Ekeland and R.Temam, Convex Analysis and Variational Problems. NorthHolland, Amsterdam, 1976.

[3] G.A.Yosifian, Some unilateral boundary value problems for elastic bodies with rugged boundaries, Preprint 99-18 (SFB 359) IWR, Universitiit Heidelberg (To appear in Trudy Seminara imeni Petrowskogo, Vol. 21 1999).

[4] G.A.Yosifian, Some homogenization problems for the system of elasticity with nonlinear boundary conditions in perforated domains, Applicable Analysis, 71, 1-4 (1999), 379-411.

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