· rendiconti del seminario matematico universita e politecnico di torino` contents m. bernardara,...

RENDICONTI

DEL SEMINARIO

MATEMATICO

Universita e Politecnico di Torino

CONTENTS

M. Bernardara,Calabi-Yau complete intersections with infinitely many lines . . 87

I. Camperi,Global hypoellipticity and Sobolev estimates for generalized SG-pseudo-differential operators. . . . . . . . . . . . . . . . . . . . . . . . 99

M. Lisi – S. Totaro,Analysis of an age-structured MSEIR model. . . . . . . . 113

H.L. Vasudeva – Mandeep Singh,Weighted Power Means of Operators. . . . . 131

N. Ujevic,An application of the montgomery identity to quadrature rules . . . 137

E. Ballico, Rank2 arithmetically Cohen-Macaulay vector bundles on certainruled surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

L. Selmani - N. Bensebaa ,An electro-viscoelastic contact problem with adhe-sion and damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Volume 66, N. 2 2008

DIRETTORE

CATTERINA DAGNINO

COMMISSIONE SCIENTIFICA (2006–08)

C. Dagnino (Direttore), R. Monaco (Vicedirettore), G. Allasia, S. Benenti, A. Collino,F. Fagnani, G. Grillo, C. Massaza, F. Previale, G. Zampieri

COMITATO DIRETTIVO (2006–08)

S. Console, S. Garbiero, G. Rossi, G. Tedeschi, D. Zambella

Proprieta letteraria riservata

Autorizzazione del Tribunale di Torino N. 2962 del 6.VI.1980Direttore Responsabile: CATTERINA DAGNINO

QUESTO FASCICOLOE STAMPATO CON IL CONTRIBUTO DI:UNIVERSITA DEGLI STUDI DI TORINO

POLITECNICO DI TORINO

Rend. Sem. Mat. Univ. Pol. Torino - Vol. 66, 2 (2008)

M. Bernardara

CALABI-YAU COMPLETE INTERSECTIONS WITH

INFINITELY MANY LINES

Abstract. We give two new examples of families of Calabi-Yau complete intersection three-folds whose generic element contains infinitely many lines.We get some results about thenormal bundles of these lines and the Hilbert scheme of lineson the threefolds. In particular,the surface swept out by such a family is not a cone.

1. Calabi-Yau complete intersections and lines on them

Throughout the paper, CY is used instead of Calabi-Yau.

The Clemens conjecture originally states that on the generic quintic threefoldthe number of rational curves in a fixed homology class is finite. More generally, theconjecture is expected to hold also for CY complete intersection threefolds in ordinaryprojective spaces (see [8]). In particular, all lines on a CYthreefold lie in the samehomology class, hence the conjecture states that the numberof lines on the genericsuch threefold is finite.

Moreover, the expected number of lines on a generic CY complete intersec-tion threefold can be computed with algebraic geometric techniques such as Schubertcalculus in the Grassmannians.

We get the same result about CY manifolds in mirror symmetry:there is a wayto predict correctly the numbernd of rational curves of a given degreed lying on thegeneric CY threefold.

Recall that a CY threefold is a complex compact Kahler threefold X with trivialcanonical bundle:

KX ≃ OX .

We will call a complete intersection of type(d1, . . . ,dk) a threefold which is a completeintersection ofk hypersurfaces inPk+3 of degreesd1, . . . ,dk respectively.

The adjunction formula for a complete intersection of type(d1, . . . ,dk)

KX∼= OX(

k

∑i=1

di −k−4)

allows to conclude that the only projective CY threefolds that are complete intersec-tions are of type(5), (the quintic threefold inP4), (3,3) and(4,2) in P5, (3,2,2) in P6

and(2,2,2,2) in P7.

Using Schubert calculus, we have the following results about the number of

87

88 M. Bernardara

lines on the generic threefold:

(5) 2875 lines(3,3) 1053 lines(4,2) 1280 lines

(3,2,2) 720 lines(2,2,2,2) 512 lines.

These results agree with mirror symmetry predictions (see [5, 6] for mirror symmetrytechniques, [12] for the case (3,3)).

The genericity assumption in Clemens conjecture is crucial: in fact we know ex-amples of CY threefolds with infinitely many lines. The simplest is the Fermat quinticthreefold inP4, defined by the equation

x50 +x5

1+x52 +x5

3 +x54 = 0.

The lines on the threefold are described in [2].

However, in [4], Clemens asked if we can find a continuous family of lines ona smooth quintic threefold which is not a cone, as in the Fermat quintic case. The firstsuch example is due to van Geemen. He found infinitely many lines on the genericthreefold of a family called the Dwork pencil. Its equation is

x50 +x5

1+x52 +x5

3+x54−5λx0x1x2x3x4 = 0

hence we get a pencil of quintic hypersurfaces inP4, whose zero fiber is the Fermat.

To see how to find lines on them, see [1], for a deeper investigation, see [14].The result is obtained by showing that on the generic threefold of the family thereare more than the expcted 2875 lines. This can be done choosing a ”good” automor-phism of the threefold and finding lines fixed by it. In this case ”good” means that itsorder does not divide the expected number of lines, so it has fixed lines. Under theaction of the automorphisms of the threefold, the orbit of one of those contains at least5000 lines, clearly more than 2875. Moreover, this example gives a positive answer toClemens [4] question. Indeed, a simple calculation of the normal bundle allows us tosay that the family of lines on the generic quintic of the pencil is not a cone.

In this paper, we are giving two new examples of families of CYthreefoldswhose generic element contains infinitely many lines. The first one is a pencil of(3,3)complete intersections, the second one a 2-parameter family of (2,2,2,2) completeintersections. Both provide a postive answer to Clemens [4]question, as stated incorollaries 1 and 2.

2. A (3,3) complete intersection pencil

In this paper we consider exclusively projective spaces over the complex fieldC.

The first example is a pencil of CY threefolds of type(3,3). On this particularpencil, we are able to construct more lines than expected (1458 instead of 1053).

Lines on Calabi-Yau threefolds 89

The equations of the generic threefoldXλ (smooth for genericλ) of the pencilare:

(1) Xλ :=

x3

0 +x31 +x3

2−3λx3x4x5 = 0

x33 +x3

4 +x35−3λx0x1x2 = 0.

This pencil is invariant under a group of automorphisms ofP5 of order 81 (see [12]).

Let φ be the involution ofP5 given by the change of coordinates(12)(45) ∈ S6,which preservesXλ. We consider its invariant subspacesV±:

(2)V+ = (a : a : b : c : c : d)V− = (q : −q : 0 : p : −p : 0).

Consider lines either contained in one of these subspaces orintersecting both; suchlines areφ-invariant. In this case there is no line lying onXλ entirely contained inV±,but we have the following result.

LEMMA 1. On the generic threefold Xλ there are 36 lines connecting the in-variant subspaces (2), hence each one is fixed byφ.

Proof. It can be easily seen that there are no points inV± lying onXλ if d = 0 orq = 0,so, without loss of generality, consider:

V+ = (a : a : b : c : c : 1)|a,b,c∈ C

V− = (1 :−1 : 0 : p : −p : 0)|p∈ C.

Lines joining such points have parametric equations:

(3) (at+s : at−s : bt : ct+ ps: ct− ps: t)

where(s : t) ranges overP1.

Substituting the equation (3) in the equations (1) ofXλ, we obtain two cubichomogeneous polynomials ins, t. The line belongs to the threefold if and only if thesepolynomials vanish identically. It appears in the following cases:

a3 =(2c3 +1)λ

12c

b =4acλ2

p2 = −2aλ

andc satisfies:64c6− (16λ6−32)c3+ λ6 = 0.

In particular, we have 6 values forc for genericλ, then we have 18 values fora and 36for p.

90 M. Bernardara

THEOREM1. On the generic threefold in the pencil Xλ there are infinitely manylines.

Proof. We know from the preceding Lemma that we have 36 lines onXλ. Pick one ofthem and call itl .

Consider the action of the group(C∗)6 on P5, where an element(a0, . . . ,a5) ∈ (C∗)6

acts componentwise by:

(a0,a1,a2,a3,a4,a5) · (x0 : x1 : x2 : x3 : x4 : x5) =

(a0x0 : a1x1 : a2x2 : a3x3 : a4x4 : a5x5)

on (x0 : . . . : x5) ∈ P5.

Let αi in (C∗)6 be the elements:

(4)α1 = (1,ω,ω−1,1,1,1)α2 = (1,1,ω,ζ,ζ,ζ−2)α3 = (1,1,1,1,ω,ω−1)

whereζ is a primitive ninth root of unity andω = ζ3.

We note that the group

G :=< α1,α2,α3 >⊂ AutXλ,

has order 81.

Two other subgroups of AutXλ are given by the actions ofS3 on the first threecoordinates and on the last three. We denote the product of these two groups byH.

The orbit of the linel under the action of the groupG has order 81, because noelement of this group fixesl .

The orbit ofl under the action ofH has 18 elements, becauseφ = (12)(45) fixesl .

Consider now the groupG×H and check that ifgh(l) = l , whereg ∈ G andh∈ H, theng(l) = l andh(l) = l (consider the points ofh(l)∩V− and then the actionof G on these points).

Hence, the order of the orbit ofl under the action of the groupG×H, is 81·18=1458. This number is larger than expected.

REMARK 1. Recall the way of counting lines proposed by S. Katz in [10]. Itis based on finding a compact moduli spaceM of the curves on the manifold, thenconstructing a rankr = dimM vector bundle with some good properties and then com-puting itsr-th Chern class. We note that, because of their constructionwith automor-phisms, our lines have the same behavior. If they were isolated, each would count asone; this would make the calculation fail. Then we deduce that each of these linesbelongs to a continuous family. Notice that this tells nothing about the number andthe geometric properties of these families, except that this excludes the case that theselines have normal bundle of the formO

P1(−1)⊕OP1(−1).


3. Normal bundle and Hilbert scheme of lines

Recall the definition of normal bundle of a lineL in a manifoldX, as the cokernel inthe exact sequence:

(5) 0−→ TL −→ TX|L −→ N −→ 0.

Now we are looking for the normal bundle of a lineL on a threefoldX, which is abundle overP1, hence we can split it as

(6) N ∼= OP1(a)⊕OP1(b).

From the CY condition we deduce

a+b= −2.

Let X be a projective variety andZ ⊂ X a subvariety. It is well known (see [11]), thatfor the Zariski tangent space to the Hilbert scheme in[Z] the following isomorphismholds:

T[Z]Hilb(X) ∼= HomX(I(Z),OZ) = HomZ(I(Z)/I(Z)2,OZ).

The right side is the zeroth cohomology group of the normal bundle ofZ in X (see [7]),thus:

(7) T[Z]Hilb(X) ∼= H0(NZ|X).

In our case, we are looking for the normal bundle of lines lying in a continuous family,hence the Zariski tangent space to the Hilbert scheme in the point corresponding tothese lines should be positive dimensional. This gives

Nl |Xλ6∼= OP1(−1)⊕O

P1(−1)

because in this case we would haveh0(N) = 0.

3.1. How to calculate the normal bundle

In this section, we show how to calculate the normal bundle ofa line on a CY completeintersection threefold and after we will apply the calculation to the lines previouslyconstructed.

Our aim is to calculatea andb in (6), trying to generalize slightly the calcula-tions in [9] to the complete intersection case.

First, letX be a hypersurface inPn andL ⊂ X a line on it. Change the coordi-nates ofPn such that the lineL has parametrization(s : t : 0 : · · · : 0); in this case, theideal IL of L is IL = (x2, · · · ,xn). Let us callFd the polynomial definingX andd itsdegree.L ⊂ X andL is the intersection of the hyperplanesx2 = · · · = xn = 0, so we canwrite:

Fd = x2F2 + · · ·+xnFn.

92 M. Bernardara

Modulo elements ofI2L, we get

Fd = x2 f2(x0,x1)+ · · ·+xn fn(x0,x1)

where eachfi is homogeneous of degreed−1; it can be seen asFi|L.

Using these exact sequences

a) 0−→ N −→ OP1(1)n−1 −→ O

P1(d) −→ 0b) 0−→ TX −→ TPn|X −→ OX(d) −→ 0

and (5), it is possible ([9]) to get the normal bundle as the kernel of the map

(8)O

P1(1)n−1 −→ OP1(d)

(s2, · · · ,sn) 7−→ ∑ni=2 fisi .

Now let X be a complete intersection of two hypersurfaces of degreed ande, givenrespectively byF = 0 andG = 0. LetL ⊂ X be parametrized as before, so we get thehomogeneous polynomialsfi andgi of degreesd−1 ande−1 respectively in(x0,x1).

From a direct calculation, we get

NL|X ≃ ker(OP1(1)n−1 M

−→ OP1(d)⊕O

P1(e))

where the map is given by the 2× (n−1) matrix M with rows given by thefi andgi .If we let A = C[x0,x1], we can rewrite this map as a mapAn−1 → A2. Hence we arelooking at the module

B = ker(An−1(1)M−→ A(d)⊕A(e))

and we know thatB has a basis of vectors of homogeneous polynomialsTi of the samedegree (within the vector)ti . In the case of a line in a threefold we havei = 1,2, hence:

N = OP1(1− t1)⊕OP1(1− t2).

In conclusion we get the following result.

THEOREM 2. Let T a vector of homogeneous polynomials of minimal degree tin (x0,x1) such that

M ·T = 0

where M is the matrix with rows given by the fi and the gi . Then the normal bundleNL|X splits in the following way:

NL|X = OP1(1)⊕O

P1(−3) if t = 0NL|X = O

P1 ⊕OP1(−2) if t = 1NL|X = O

P1(−1)⊕OP1(−1) otherwise.

Proof. This follows easily from the above considerations, remembering that we shouldhave, for the CY condition,t1−1+ t2−1 = −2.

The argument is essentially the same for a generic CY complete intersectionthreefold in projective space.


3.2. The normal bundle of constructed lines

We calculate the normal bundle of the lines constructed in the previous section on thegeneric threefoldXλ.

LEMMA 2. Let λ be generic and l⊂ Xλ be the line parametrized by

(at+s : at−s : bt : ct+ ps: ct− ps: t)

as in Lemma 1. Then its normal bundle on Xλ splits as:

Nl |Xλ∼= OP1 ⊕OP1(−2).

Proof. Recall thatNl |Xλ

6∼= OP1(−1)⊕OP1(−1).

Define new coordinates:

x0 = y0 +ay5

x1 = −y0 +y1 +ay5

x2 = y2 +by5

x3 = py0 +y3 +cy5

x4 = −py0 +y4+cy5

x5 = y5.

l has now parametrization(s : 0 : 0 : 0 : 0 :t).We can obtain the matrixM, with coefficients homogeneous quadratic polynomials iny0 andy5:

((y0−ay5)

2 b2y25 λp(y0y5)−cλy2

5 −λp(y0y5)−cλy25

−bλ(y0y5 +y25) λ(y2

0−a2y25) (py0 +cy5)

2 (py0−cy5)2

).

Now we verify that there are no nonzero vectorsB∈ C4 such thatM ·B= 0. This leads

to:Nl |Xλ 6

∼= OP1(1)⊕OP1(−3).

COROLLARY 1. Let l be a line on Xλ in the family constructed in section 2.Then its normal bundle splits as:

Nl |Xλ∼= OP1 ⊕OP1(−2).

This means that the surface swept out by these lines on Xλ is not a cone.

Proof. Let l be as in Lemma 2. Each line constructed in Theorem 1 can be obtained byl using an automorphism ofXλ.

Moreover we can conclude the dimension of the Hilbert schemeis positive, inparticular

dimT[l ]Hλ = h0(Nl |Xλ) = 1

for the linesl we constructed in section 2.

94 M. Bernardara

4. A (2,2,2,2) two-parameter Family

We now give a new example, a two-parameter family of(2,2,2,2) threefolds inP7.Consider the family (smooth for generic (λ,µ)) obtained by the complete intersectionof the four quadrics

(9) Xλ,µ :=

x20 +x2

1+x22 +x2

3+x24 +x2

5 −2µx6x7 = 0

x20 +x2

1+x22 +x2

3 +x26 +x2

7−2λx4x5 = 0

x20 +x2

1 +x24 +x2

5+x26 +x2

7−2λx2x3 = 0

x22 +x2

3+x24 +x2

5+x26 +x2

7−2λx0x1 = 0.

As in the previous case, on the generic threefold more than the 512 expected lines areshown.The technique is the same: in this case we takeφ to be the order 3 automorphism ofXλ,µ given by the permutation of coordinates(135)(246) in P7. OnP7 we consider itsinvariant subspaces:

V+ = (a : b : a : b : a : b : c : d)

Vω = (p : q : ωp : ωq : ω2p : ω2q : 0 : 0)

where(a : b : c : d) ∈ P3, (p : q) ∈ P1 andω ∈ C is primitive third root of the unity.

LEMMA 3. For generic(λ,µ), there are 8 lines on Xλ,µ intersecting both V+and Vω.

Proof. First we verify that the points withb = 0 and the ones withp = 0 don’t lie onthe threefold, hence we can consider:

(10) V+ = (a : 1 : a : 1 : a : 1 : c : d)|a,c,d ∈ C

(11) Vω = (1 : q : ω : ωq : ω2 : ω2q : 0 : 0)|q∈ C.

Lines joining these points have parametric equation

(12) (s+at : qs+ t : ωs+at : ωqs+ t : ω2s+at : ω2qs+ t : ct : dt)

where(s : t) ∈ P1.Substituting these values into the equations of the generic threefold,the line lies onXλ,µ if and only if

a = −λ +qλq+1

d =3a2+3

2µc

whereq is a root ofq2 +2λq+1= 0


andc is a root of

4µ2c4 +(2a2−2λa+2)c2+(3a2+3)2 = 0.

We get the proof, remarking that if we have 2 values fora, then it is clear that we have8 values forc for the generic couple(λ,µ) and that this does not depend ona and onq.

THEOREM 3. On the generic threefold of the pencil Xλ,µ there are infinitelymany lines.

Proof. A subgroup of AutXλ,µ is given by the action ofS3 on the first three pairs ofcoordinates. The orbit of one of the constructed lines underthis automorphism groupconsists of 2 lines, becauseφ belongs to this group.Another subgroup of automorphisms is generated by the permutations of coordinates(12), (34), (56) and(78): the constructed lines are not fixed by any of these automor-phisms, hence the orbit of each line under the action of this subgroup has 16 elements.Let G be the product of these two groups (in particular,G≤ S8).Let H be the subgroup of(C∗)8, acting onP

7 by the coordinatewise product, withgenerators

α1 = ( −1, −1, 1, 1, 1, 1, 1, 1)α2 = ( 1, 1, −1, −1, 1, 1, 1, 1)α3 = ( 1, 1, 1, 1, −1, −1, 1, 1).

The orbit of each line under its action consists of 8 lines.Consider now the groupG×H: we have to check that ifgh(l) = l , whereg∈ G andh∈ H, theng(l) = l andh(l) = l (consider the points in the seth(l)∩Vω and then theaction ofG on these points). We get finally that the orbit of each line under the actionof this group consists of 256 elements.The key remark now is that the orbits of the lines are disjoint, and this is made making atable comparing the values obtained for the points inVω andV+ starting from differentvalues ofc.We finally get onXλ,µ at least 2048 lines, that is more than expected.

REMARK 2. The same argument used in Remark 1 is valid in this case, so allthe constructed lines belong to a continuous family.

4.1. The normal bundle

LEMMA 4. The line l⊂ Xλ,µ parametrized by

(s+at : qs+ t : ωs+at : ωqs+ t : ω2s+at : ω2qs+ t : ct : dt)

as in Lemma 3, has normal bundle

Nl |Xλ,µ∼= OP1 ⊕OP1(−2).

96 M. Bernardara

Proof. As before:Nl |Xλ,µ

6∼= OP1(−1)⊕OP1(−1).

Now with the change of coordinates:

x0 = y0 +ay7

x1 = qy0 +y1+y7

x2 = ωy0 +y2+ay7

x3 = ωqy0 +y3+y7

x4 = ω2y0 +y4 +ay7

x5 = ω2qy0 +y5+y7

x6 = y6 +cy7

x7 = dy7

the linel gets parametrization(s : 0 : 0 : 0 : 0 : 0 : 0 :t). We calculate the matrixM, withcoefficients linear homogeneous polynomials iny0 andy7:

qy0 +y7 ωy0 +ay7 ωqy0 +y7 ω2y0 +ay7 ω2qy0 +y7 −µdy7qy0 +y7 ωy0 +ay7 ωqy0 +y7 −λ(ω2qy0 +y7) −λ(ω2y0 +ay7) cy7qy0 +y7 −λ(ωqy0 +y7) −λ(ωy0 +ay7) ω2y0 +ay7 ω2qy0 +y7 cy7

−λ(y0 +ay7) ωy0 +ay7 ωqy0 +y7 ω2y0 +ay7 ω2qy0 +y7 cy7

.

We now verify that for generic(λ,µ) there are no nonzero vectorsB in C6 such thatM ·B = 0 and thenNl |Xλ,µ

6∼= OP1(1)⊕OP1(−3).

COROLLARY 2. Let l be a line on Xλ,µ in the family previously constructed.Then its normal bundle splits as:

Nl |Xλ,µ∼= OP1 ⊕OP1(−2).

This means that the surface swept out by these lines on Xλ,µ is not a cone.

Moreover, the dimension of the Hilbert scheme is positive, in particular:

dimT[l ]Hλ = h0(Nl |X(λ,µ)) = 1

for the linesl we constructed.

References

[1] A LBANO A. AND KATZ S.Van Geemen’s of lines on special quintic threefolds, Manuscripta Math.70(1991), 183–188.

[2] A LBANO A. AND KATZ S., Lines on the Fermat quintic threefold and the infinitesimal generalizedHodge conjecture, Trans. AMS324(1991), 353–368.

[3] CLEMENS H., Curves on higher-dimensional complex projective manifolds, in: “Proceedings of theinternational congress of mathematicians”, vol. 1, 2 (Berkley, CA, 1986), 634–640, AMS ProvidenceRI, 1987.

[4] CLEMENS H., Problems on 3-folds with trivial canonical bundle, in: “Birational geometry of alge-braic varieties, open problems”, The XXIIIrd Internaitonal Symposium, Division of Mathematics, TheTaniguchi Foundation, 1988.


[5] COX D. A. AND KATZ S.,Mirror symmetry and algebraic geometry, Math. surveys and monographs68, AMS Providence RI, 1999.

[6] GROSSM., HUYBRECHTSD., JOYCE D., Calabi-Yau Manifolds and related geometry, Universitext,Springer Verlag, Berlin Heidelberg, 2003.

[7] HARTSHORNER.,Algebraic geometry, Graduate Texts in Mathematics52, Springer-Verlag, New YorkNY, 1977.

[8] JOHNSEN T. AND KNUTSEN A. L., Rational curves in Calabi-Yau threefolds, Communications inalgebra31 (8)(2003), 3917–53.

[9] K ATZ S.,On the finiteness of rational curves on quintic threefolds, Compositio Math.60 (1986), 151–162.

[10] KATZ S., Rational curves on Calabi-Yau threefolds, in: “Essays on mirror manifolds” (S.-T. Yaueditor), 168–180. International Press, Hong Kong, 1992.

[11] KOLLAR J.,Rational curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebi-ete, 3. Folge, B. 32, Springer Verlag, Berlin Heidelberg, 1996.

[12] L IBGOBER A. AND TEITELBAUM J., Lines on Calabi-Yau complete intersections, mirror symmetryand Picard-Fuchs Equations, Internat. Math. Res. Notices (1993), 29–39.

[13] MORRISOND.R.,Mirror symmetry and rational curves on quintic threefolds:a guide for mathemati-cians, Journal of AMS6 (1993), 223–247.

[14] MUSTATA A., Degeree one curves in the Dwork pencil and the mirror quintic, arXiv:math.AG/0311252.

AMS Subject Classification: 14J32, 14J30.

Marcello BERNARDARA, Laboratoire J. A. Dieudonne, Universite de Nice Sophia Antipolis, ParcValrose, 06108 Nice Cedex 2, FRANCEe-mail: [email protected]

Lavoro pervenuto in redazione il 03.10.2006 e, in forma definitiva, il 11.09.2007.


I. Camperi

GLOBAL HYPOELLIPTICITY AND SOBOLEV ESTIMATES

FOR GENERALIZED SG-PSEUDO-DIFFERENTIAL

OPERATORS

Abstract. We prove global hypoellipticity and maximal Sobolev estimates for a class ofgeneralized SG-pseudo-differential operators. Applications are given to partial differentialoperators with polynomial coefficients inRn.

1. Introduction

In this paper we consider linear partial differential operators or pseudo-differential op-eratorsP = p(x,D) globally defined inRn, and we study for them the problem of theglobal hypoellipticity. Namely, writingS (Rn) for the Schwartz space of the rapidlydecaying functions andS ′(Rn) for its dual, we assume thatP : S (Rn)→ S (Rn) extendsto a mapS ′(Rn) → S ′(Rn). We then recall thatP is globally hypoelliptic if, for anygiven f ∈ S (Rn), all the solutionsu∈ S ′(Rn) of the equationPu= f belong toS (Rn).This definition does not imply the local regularity of the solutions ofP, however itis natural to limit attention to operatorsP which are already known to be hypoellip-tic in the Schwartz sense, i.e.Pu∈ C∞(Ω) for u ∈ D ′(Ω) implies u ∈ C∞(Ω), foreveryΩ ⊂ R

n. Relevant classes of globally hypoelliptic operators wereidentified byShubin [10], taking as basic example the harmonic oscillator of Quantum MechanicsP = −∆+ |x|2. A generalization of the classes of Shubin was then given by Boggiatto,Buzano, Rodino [3]. In a somewhat different direction, Parenti [9] and Cordes [5] in-troduced the so-calledSG-classes. In the present paper we presentSGm

ρ,δ extensions ofSG-classes, allowing new applications to global hypoellipticity.

Basic example, where to test the previous contributions andour results, is givenby partial differential operators with polynomial coefficients inRn

(1) P = ∑cα,βxαDβ (Dxj = −i∂xj )

where the coefficientscα,β are complex numbers, and in the sumα,β run over a finiteset of indices. Consider the symbol

p(x,ξ) = ∑cα,βxαξβ.

The Newton’s polyhedronN p of the polynomialp(x,ξ) is defined as the convex hullin R2n of all multi-indices(α,β) ∈ N2n, with cα,β 6= 0, and the origin (see for exampleGindikin, Volevich [7]). We then associate top(x,ξ) the function onR2n

λ(x,ξ) = ∑(α,β)∈N p

∣∣∣xαξβ∣∣∣

99

100 I. Camperi

and we say thatP, or p(x,ξ), is λ-elliptic if

(2) cλ(x,ξ) ≤ |p(x,ξ)| ≤Cλ(x,ξ)

for positive constantsc andC and large|x|+ |ξ|. Theλ-ellipticity grants the globalellipticity of P, if N p satisfies suitable geometric properties. As basic example ofpolyhedron having a ”good geometry” we have from Shubin [10]the simplex

(3) N p = (α,β), |α|+ |β| ≤ m

wherem is a fixed positive integer, and we letα,β run here inR2n. In Boggiatto,Buzano, Rodino [3], as generalization of (3), the Newton’s polyhedronN p was as-sumed complete, this means that it does not contain sides parallel to coordinate axesand the outer normals to the non-coordinate faces have strictly positive components. IntheSG-case of Parenti [9] and Cordes [5] we have

(4) N p = (α,β), |α| ≤ m2, |β| ≤ m1

wherem1 ≥ 0, m2 ≥ 0 are fixed integers. Note that the polyhedron in (4) is not com-plete, since the outer normals to the non-coordinate faces contain zero components.Hence the results of Boggiatto, Buzano, Rodino [3] cannot beapplied, nevertheless thecorrespondingλ-elliptic operators are globally hypoelliptic. In the present paper weshall go further with respect to theSG-case, considering for a fixedγ, 0< γ < 1:

(5) P = ∑0≤β≤m10≤α≤γβ

cα,βxαDβ

where for simplicity we assumeα∈N, β∈N, i.e. P is an ordinary differential operator.Let us assumem2 = γm1 is an integer. The corresponding Newton’s polyhedron inR2

(6) N p = (α,β), 0≤ α ≤ γβ, 0≤ β ≤ m1

has a face, namely the segment connecting(0,0) and(m2,m1), with outer normal con-taining a negative component. Theλ-ellipticity condition (2) can be written in a sim-plified form:

(7) c(1+ |ξm1|+ |xγm1ξm1|) ≤ |p(x,ξ)| ≤C(1+ |ξm1|+ |xγm1ξm1|) .

As an application of our pseudo-differential calculus, we shall obtain thatP in (5) isglobally hypoelliptic, under the assumption (7).

Consider as an example

P =(1+x2)D4 +1

with symbolp(x,ξ) =

(1+x2)ξ4 +1

Global hypoellipticity and Sobolev estimates 101

of the form (5) withm1 = 4, γ = 1/2. The condition (7) is obviously satisfied. Ourresult cannot be extended to the case whenγ ≥ 1 in (5). As counter-example, considerthe operator

(8) P =(1+x2)D2 +2.

The symbolp(x,ξ) =(1+x2

)ξ2 +2 is λ-elliptic with respect to

N p = (α,β), 0≤ α ≤ β, 0≤ β ≤ 2 ,

that is we are in the case (6), butγ = 1. The corresponding homogeneous equation

Pu= −(1+x2)u′′ +2u= 0

admits the solutionu(x) = 1+x2 ∈ S ′(R),

which does not belong toS (R). ThereforeP in (8) is not globally hypoelliptic. Letus conclude by observing that the characterization of the polyhedronsN p for whichλ-ellipticity grants global hypoellipticity, as well as thegeneral characterization of theoperators with polynomial coefficients (1) which are globally hypoelliptic, are widelyopen problems.

The contents of the paper are the following. In the next Section 2 we presentour SGm

ρ,δ-pseudo-differential calculus. It is more general than thecalculus of Cordes[5] and it does not enter the results of Beals [1]. However, itcan be seen as a particularcase of the so-called Weyl-Hormander calculus, see Hormander [6]. For this reasonwe omit the proofs, and emphasize only the peculiarities of the construction of theparametrices.

In the last part of this work, Section 4, we focus our attention to SGSobolevspacesHs of Cordes [5]: we prove that an operator with symbol inSGm

ρ,δ is continuous

from Hs into Hs−m. Finally, we define a Sobolev spaceH∗ modelled on (5) and weshow a maximal inequality that tie upH∗-norm with L2-norm plus a Sobolev norm.To prove this assertion, proporties of Newton’s polyhedronare fundamental. As aconsequence of the maximal estimate, we obtain the Fredholmproperty of the mapP : H∗ → L2.

Recently Cappiello, Gramchev, Rodino [4] have proved that,in the case whenthe symbols of the pseudo-differential operatorsSGm

ρ,δ satisfy additional analytic bounds,the regularity inS (Rn) of the solution can be replaced by regularity in Gelfand-Shilovclasses, expressing holomorphic extension and exponential decay. Via our results inSection 3, it is possible to apply the results of Cappiello, Gramchev, Rodino [4] to theoperators with polynomial coefficients (5).

2. SGmρ,δ operators

We introduce some notational conventions. Letm= (m1,m2)∈R2, and letρ1,ρ2,δ1,δ2

be real numbers with 0≤ δ j < ρ j ≤ 1, j = 1,2, and denoteρ = (ρ1,ρ2), δ = (δ1,δ2).Let e1 = (1,0), e2 = (0,1), e= (1,1).

102 I. Camperi

We define pseudo-differentialoperators of classSGmρ,δ and investigate their basic

properties.

DEFINITION 1. We say that a function p(x,ξ) ∈ C∞(R2n) is a symbol of classSGm

ρ,δ if there exists a positive constant Cα,β such that

∣∣∣Dαξ Dβ

x p(x,ξ)∣∣∣≤Cα,β〈ξ〉

m1−ρ1|α|+δ1|β|〈x〉m2−ρ2|β|+δ2|α|

for every(x,ξ) ∈ R2n andα,β ∈ Nn, where〈ξ〉 = (1+ |ξ|2)1/2

and〈x〉 = (1+ |x|2)1/2

.

Some classes were treated by Cordes [5] under the assumptionδ2 = 0.

We also define

SG−∞ =\

m∈R2

SGmρ,δ = S (R2n), SG∞ =

[

m∈R2

SGmρ,δ,

whereS (R2n) is the Schwartz space.

For p(x,ξ) ∈ SGmρ,δ we define the semi-normsNα,β(p), by

Nα,β(p) = sup(x,ξ)∈R2n

〈ξ〉−m1+ρ1|α|−δ1|β|〈x〉−m2+ρ2|β|−δ2|α|∣∣∣Dα

ξ Dβx p(x,ξ)

∣∣∣ .

Then SGmρ,δ is a Frechet space with respect to the topology induced by these semi-

norms.It is easy to prove that the following relations hold:

1. If mj ≤ m′j , ρ′

j ≤ ρ j , δ j ≤ δ′j , j = 1,2, thenSGmρ,δ ⊆ SGm′

ρ′,δ′ .

2. If p∈ SGmρ,δ, thenDα

ξ Dβx p(x,ξ) ∈ SGm′′

ρ,δ with

m′′ = m− (ρ1 |α|− δ1 |β|)e1− (ρ2 |β|− δ2 |α|)e2.

Corresponding top(x,ξ)∈SGmρ,δ, we define the pseudo-differentialoperatorP= p(x,D)

by the standard formula

(9) Pu(x) = p(x,D)u(x) = (2π)−nZ

Rnei〈x,ξ〉p(x,ξ)u(ξ)dξ, u∈ S (Rn),

whereu is the Fourier transform.We shall denote byOPSGm

ρ,δ the space of all operators of the form (9), with symbols inSGm

ρ,δ.

For a symbolp ∈ SGm0

ρ,δ and a sequence of symbolsp j , j ∈ N, p j ∈ SGmj

ρ,δ, mj =

(mj1,m

j2), m0

k > m1k > .. . > mj

k > .. . ,mjk →−∞, as j → ∞, k = 1,2, we will say thatp


has the asymptotic expansion∞

∑j=0

p j , written asp∼∞

∑j=0

p j , if for everyN ∈ N we have

p−N−1

∑j=0

p j ∈ SGmN

ρ,δ.

The following theorems can be proved making simple changes to the proofs inCordes [5], in Parenti [9] or else by applying the general Weyl-Hormander calculus in[6], so we omit the proofs.

THEOREM 1. Given p∈ SGmρ,δ, the operator P defined by (9) is linear and

continuous from the Schwartz spaceS into itself. Furthermore, P can be extended to alinear and continuous map fromS ′ into itself.

THEOREM 2. Let pj ∈ SGmj

ρ,δ, j ∈ N, where mj = (mj1,m

j2,), with

m0k > m1

k > m2k > .. . , mj

k →−∞, as j→ ∞, k = 1,2.

There exists a symbol p∈ SGm0

ρ,δ such that

p∼∞

∑j=0

p j .

THEOREM3. Let P= p(x,D) ∈ OPSGm1

ρ,δ, Q= q(x,D) ∈ OPSGm2

ρ,δ. Then, there

exists a symbol r∈ SGm1+m2

ρ,δ such that q(x,D)p(x,D) = r(x,D). Furthermore

r(x,ξ) ∼ ∑α

1α!

∂αξ q(x,ξ)Dα

x p(x,ξ).

DEFINITION 2. A symbol p∈ SGmρ,δ is said to be hypoelliptic if there exist

R,C0,C0,α,β > 0 and m′ = (m′1,m

′2) ∈ R

2 such that

(10) |p(x,ξ)| ≥C0〈ξ〉m′1〈x〉m′

2

and

(11)∣∣∣Dα

ξ Dβx p(x,ξ)

∣∣∣≤C0,α,β |p(x,ξ)| 〈ξ〉−ρ1|α|+δ1|β|〈x〉−ρ2|β|+δ2|α|

for all α,β ∈ Nn and(x,ξ) ∈ R2n with |x|+ |ξ| > R.

DEFINITION 3. A symbol p∈ SGmρ,δ is called elliptic if it satisfies(10) with

m′ = m.

Let us observe that (10) withm′ = m implies (11); therefore an elliptic symbolis also hypoelliptic.

We say thatQ is a parametrix ofP if we haveQP− I ∈ OPSG−∞ andPQ−I ∈ OPSG−∞. We speak of a left (right) parametrixQ of P if only the second (first)condition holds.

104 I. Camperi

THEOREM4. Let P be an operator with a hypoelliptic symbol p∈ SGmρ,δ. Then,

it admits a parametrix Q with symbol q∈ SG−m′

ρ,δ .

Proof. As standard we define a sequence of symbolsq j ( j ≥ 1) inductively by

(12)

q0(x,ξ) =1

p′(x,ξ)

q j(x,ξ) = −

∑|γ|= j−k

k< j

1γ!

∂γξqkD

γxp′

q0, j = 1,2, . . . ,

for every(x,ξ) ∈ R2n, wherep′(x,ξ) = p(x,ξ) for |x|+ |ξ| ≥ R, p′(x,ξ) ∈ C∞(R2n)

with p′(x,ξ) 6= 0. It is easy to prove thatq j ∈ SG−m′−(ρ−δ) jeρ,δ by using (10), (11). Then,

by Theorem 2 there exists an operatorQ = q(x,D) ∈ OPSG−m′

ρ,δ with symbolq(x,ξ) ∼∞

∑j=0

q j(x,ξ). Applying Theorem 3 we can easy verify thatQ is a parametrix ofP.

THEOREM 5. Let p be a hypoelliptic symbol in SGmρ,δ and let f∈ S . If u ∈ S ′

is a solution of the corresponding equation Pu= f , then u∈ S . That is, P is globallyhypoelliptic.

Proof. By Theorem 4 there exist an operatorQ with symbol inSG−m′

ρ,δ and an operator

R with symbol inSG−∞ = S(R2n)

such that

Q f = QPu= u+Ru.

We haveu = Q f −Ru.

Since a pseudo-differential operator with symbol inS(R2n)

is regularizing, i.e. it mapsS ′(R2n)

into S(R2n), thenu∈ S .

EXAMPLE 1. The symbolp(x,ξ) = 〈x〉γξ+ i, with (x,ξ) ∈ R2 is hypoelliptic as

symbol inSG(1,γ)(1,1),(0,γ) if 0 ≤ γ < 1. In fact, the condition (10) is fulfilled form′ = (1,0).

Moreover,

∣∣Dξ p(x,ξ)∣∣= 〈x〉γ =

1+ |ξ|1+ |ξ|

〈x〉γ ≤|p(x,ξ)|

1+ |ξ|〈x〉γ ∼ |p(x,ξ)| 〈x〉γ〈ξ〉−1

and

|Dxp(x,ξ)| ∼ |x| 〈x〉γ−2 |ξ| =|x|

〈x〉2 〈x〉γ |ξ| ≤

|x|

〈x〉2

(〈x〉γ |ξ|+1

)

∼|x|

〈x〉2 |p(x,ξ)| ≤|x|+1

〈x〉2 |p(x,ξ)| ∼ 〈x〉−1 |p(x,ξ)|


when|x|+ |ξ| is large. It is easy to verify that also higher order derivatives satisfy (11).

3. Newton’s polyhedron

Consider a polynomialp(x,ξ) = ∑cα,βxαξβ

in the variable(x,ξ) ∈ R2, with constant complex coefficientscα,β ∈ C.

Let R2+ be the positive quadrant in the plane:

R2+ =

(α,β) ∈ R

2,α ≥ 0,β ≥ 0

.

DEFINITION 4. Let p(x,ξ) be defined as before and consider the finite set ofpoints ofR2

+ A =(α,β) ∈ R2

+,cα,β 6= 0

. The Newton’s polyhedronN p of the poly-nomial p(x,ξ) is the convex hull inR2

+ of A∪0 .

We shall deal with polynomials in two variables of type:

(13) p(x,ξ) = ∑0≤β≤m10≤α≤γβ

cα,βxαξβ

with 0 < γ < 1. We define

λ(x,ξ) = 1+ |ξm1|+ |xγm1ξm1| ∼√

1+ ξ2m1 +x2γm1ξ2m1 ∼ ∑0≤β≤m10≤α≤γβ

∣∣∣xαξβ∣∣∣

and we suppose that there exists a constantC > 0 such that

|p(x,ξ)| ≥Cλ(x,ξ).

We then say that the polynomial (13) satisfies the condition of λ-ellipticity. Under thesehypotheses it is obvious that|p(x,ξ)| ∼ λ(x,ξ) andC0,m1 6= 0,Cγm1,m1 6= 0.

Hence the Newton’s polyhedronN p of the polynomial (13), given by (6), is aright triangle with verticesV p = (0,0),(0,m1),(γm1,m1) .

We can prove the following result:

PROPOSITION1. If the symbol(13)is λ-elliptic andγ < 1, then it is hypoelliptic

in SG(m1,γm1)(1,1),(0,γ).

To prove Proposition 1, we need a preliminary result.

LEMMA 1. Given x∈(R

+0

)k, a finite subset A⊂

(R

+0

)kand a convex linear

combinationβ = ∑α∈A

cαα, we have that

xβ ≤ ∑α∈A

cαxα.

106 I. Camperi

A proof of this lemma can be found in Boggiatto-Buzano-Rodino [3].

Proof of Proposition 1.It is obvius that the condition (10) is satisfied withm′ = (m1,0).Now we prove that the polynomial satysfies the condition (11)for the derivative of or-der one, the others inequalities follow immediately. We have

|Dxp(x,ξ)| =

∣∣∣∣∣∣∣∣∑

0≤β≤m10<α≤γβ

αcα,βxα−1ξβ

∣∣∣∣∣∣∣∣

1+ |x|1+ |x|

∼ 〈x〉−1

∣∣∣∣∣∣∣∣∑


αcα,βxα−1ξβ (1+ |x|)

∣∣∣∣∣∣∣∣

≤ 〈x〉−1

∑


α∣∣cα,β

∣∣∣∣∣xα−1ξβ

∣∣∣+ ∑0≤β≤m10<α≤γβ

α∣∣cα,β

∣∣∣∣∣xαξβ

∣∣∣

.

Every term∣∣xα−1ξβ∣∣ is such that(α−1,β) ∈ N p. Using Lemma 1 and theλ-

ellipticity of p(x,ξ) we obtain∣∣∣xα−1ξβ

∣∣∣≤ λ(x,ξ) ≤C|p(x,ξ)| .

Applying the same argument to the terms∣∣xαξβ∣∣ , we have

∣∣∣xαξβ∣∣∣≤ λ(x,ξ) ≤ |p(x,ξ)| .

Therefore we can conclude that:

|Dxp(x,ξ)| ≤C|p(x,ξ)| 〈x〉−1.

With analogous reasonings we have

∣∣Dξ p(x,ξ)∣∣=

∣∣∣∣∣∣∣∣|x|γ ∑

0<β≤m10≤α≤γβ

βcα,βxα|x|−γξβ−1

∣∣∣∣∣∣∣∣

1+ |ξ|1+ |ξ|

≤(1+ |x|γ

)∑


β∣∣cα,β

∣∣ |x|α−γ|ξ|β−11+ |ξ|1+ |ξ|

∼ 〈x〉γ〈ξ〉−1

∑


β∣∣cα,β

∣∣ |x|α−γ|ξ|β−1+ ∑0<β≤m10≤α≤γβ

β∣∣cα,β

∣∣ |x|α−γ|ξ|β

.


We have take out the term|x|γ from the sum, therefore all the terms contained in thetwo sums are associated to a point that belongs to the Newton’s polyhedronN p and wecan apply the same argument used before. So we conclude that

∣∣Dξ p(x,ξ)∣∣≤C|P(x,ξ)| 〈x〉γ〈ξ〉−1.

4. Sobolev estimates

DEFINITION 5. The Sobolev space Hs, s= (s1,s2) ∈ R2 is defined by

Hs =

u∈ S ′/〈x〉s2〈D〉s1u∈ L2(Rn)

.

The Sobolev spaceHs is a Banach space with the norm

(14) ‖u‖Hs = ‖〈x〉s2〈D〉s1u‖L2 , u∈ Hs.

In particular the spaceHs is a Hilbert space with the inner product

(u,v)Hs = (〈x〉s2〈D〉s1u,〈x〉s2〈D〉s1v)L2, u,v∈ Hs.

Note that the pseudo-differential operator〈x〉s2〈D〉s1 ∈ OPSGsρ,δ is invertible, as an op-

erator fromS into itself (or fromS ′ into itself), with inverse〈D〉−s1〈x〉−s2 ∈ OPSG−sρ,δ.

In particular〈x〉s2〈D〉s1 is elliptic of orders.From (14) we conclude that〈x〉s2〈D〉s1 is an isometry fromHs into L2.

We now prove the boundedness of theOPSGmρ,δ operators on the spacesHs,

following the lines of Parenti [9], Cordes [5].

PROPOSITION2. Given p(x,ξ) ∈ SGmρ,δ, the operator P defined by(9) is con-

tinuous from Hs into Hs−m if and only if the operator

〈x〉s2−m2〈D〉s1−m1P〈D〉−s1〈x〉−s2

is continuous from L2 into L2.

Proof. Let u∈ Hs. There existsv∈ L2 such that

(15) u = 〈D〉−s1〈x〉−s2v

and we have‖u‖Hs =

∥∥〈D〉−s1〈x〉−s2v∥∥

Hs = ‖v‖L2.

Using (15) and (14), we obtain

‖Pu‖Hs−m =∥∥〈x〉s2−m2〈D〉s1−m1Pu

∥∥L2

=∥∥〈x〉s2−m2〈D〉s1−m1P〈D〉−s1〈x〉−s2v

∥∥L2.

108 I. Camperi

We can conclude that

‖Pu‖Hs−m ≤C‖u‖Hs, u∈ Hs

if and only if

∥∥〈x〉s2−m2〈D〉s1−m1P〈D〉−s1〈x〉−s2v∥∥

L2 ≤C‖v‖L2, v∈ L2.

PROPOSITION3. Let P= p(x,D) be an operator in OPSGmρ,δ. Then Q= q(x,D)=

〈x〉s2−m2〈D〉s1−m1P〈D〉−s1〈x〉−s2 is in OPSG(0,0)ρ,δ .

This is an obvious consequence of Theorem 3.

PROPOSITION4. Let p(x,ξ) ∈ SG(0,0)ρ,δ , then the associated pseudo-differential

operator P is continuous from L2 into itself.

TheL2-boundedness of operators with symbol inSG(0,0)ρ,δ was proved by Cordes

[5] when δ2 = 0, see also Beals [1], Beals-Fefferman [2]. In the present paper themost relevant case is represented by the caseδ2 > 0, which remained unexplored in theabove mentioned articles. However, Proposition 4 can be seen now as a consequence ofthe results in Hormander [6]. In factSGm

ρ,δ classes are included in the Weyl-Hormander

calculus and actually we getL2-boundedness also for 0≤ δ j ≤ ρ j ≤ 1, δ j 6= 1, j = 1,2,m= 0.

THEOREM 6. Given a symbol p(x,ξ) ∈ SGmρ,δ, the operator P defined by(9) is

continuous from Hs into Hs−m.

Proof. The proof is an obvious conseguence of Proposition 2, Proposition 3, Proposi-tion 4.

Applying Theorem 6 to the parametrixQ∈ OPSG−m′

ρ,δ in Theorem 4 and writing

u = QPu−Ruas in the proof of Theorem 5 we get for anyt ∈ R2 the estimate

‖u‖m′ ≤C(‖Pu‖L2 +‖u‖Ht ) , u∈ L2.

So for example for the operator with polynomial coefficientsP in Section 3, in theproof of Proposition 1 we hadm′ = (m1,0), so we obtain

∑0≤β≤m1

∥∥∥Dβu∥∥∥

L2≤C(‖Pu‖L2 +‖u‖Ht ) .

In the following we shall improve this estimates by adding inthe sum in the left-handside all the other terms

∥∥xαDβu∥∥

L2 with 0 < α ≤ γβ.


LEMMA 2. Let p(x,ξ) be the symbol(13). If p(x,ξ) is hypoelliptic and q(x,ξ)

is the symbol of its parametrix with asymptotic expansion∞

∑l=0

ql (x,ξ), cf. the proof of

Theorem 4, then the following inequality holds:

(16)∣∣∣Dα

ξ Dβxql

∣∣∣≤Cα,β |q0(x,ξ)| 〈ξ〉−(α+l)〈x〉−(β+l)+γ(α+l), l = 0,1,2, . . . .

Proof. First we prove that the inequality holds forq1. Using (12) and Leibniz’ formulawe obtain∣∣∣Dα

ξ Dβxq1(x,ξ)

∣∣∣=∣∣∣Dα

ξ Dβx

(∂ξq0(x,ξ)Dxp(x,ξ)q0(x,ξ)

)∣∣∣

≤ ∑α1+α2=αα3+α4=α2

∑β1+β2=ββ3+β4=β2

(αα1

)(α2

α3

)(ββ1

)(β2

β3

)∣∣∣Dα1ξ ∂ξDβ1

x q0(x,ξ)∣∣∣

·∣∣∣Dα3

ξ Dβ3+1x p(x,ξ)

∣∣∣∣∣∣Dα4

ξ Dβ4x q0(x,ξ)

∣∣∣ .

Using hypoellipticity of the symbolsq0(x,ξ) andp(x,ξ), we can deduce (16). Now wesuppose that inequality (16) is true untill and we prove that it holds forl + 1. Usingthe same arguments, we obtain

∣∣∣Dαξ Dβ

xql+1(x,ξ)∣∣∣

=

∣∣∣∣∣Dαξ Dβ

x

(l

∑j=0

1(l +1− j)!

∂l+1− jξ q j(x,ξ)Dl+1− j

x p(x,ξ)q0(x,ξ)

)∣∣∣∣∣

≤l

∑j=0

1(l +1− j)! ∑

α1+α2=αα3+α4=α2

∑β1+β2=ββ3+β4=β2

(αα1

)(α2

α3

)(ββ1

)(β2

β3

)

·∣∣∣Dα1

ξ ∂l+1− jξ Dβ1

x q j(x,ξ)∣∣∣∣∣∣Dα3

ξ Dβ3+l+1− jx p(x,ξ)

∣∣∣∣∣∣Dα4

ξ Dβ4x q0(x,ξ)

∣∣∣ .

Using inductive hypothesis on the first term and hypoellipticity of p(x,ξ) andq(x,ξ)respectively to the others two terms, we easily obtain the conclusion.

PROPOSITION5. Let P be the operator with symbol(13) and Q its parame-trix. Then xαDβQ is continuous from L2 into itself, for every(α,β) in the Newton’spolyhedronN p of p(x,ξ).

By applying the composition Theorem 3, sinceQ∈ SG(−m1,0)ρ,δ , xα ∈ SG(0,γm1)

ρ,δ ,

ξβ ∈ SG(m1,0)ρ,δ for (α,β) ∈ N p, we obtainxαξβQ ∈ OPSG(0,γm1)

ρ,δ , that does not grant

L2-boundedness. A more refined argument is therefore necessary.

Proof. First we prove thatxαξβql (x,ξ) ∈ SG(−l ,−l+γl)(1,1),(0,γ) , l = 0,1,2, . . . .

110 I. Camperi

By Leibniz’ formula we have

(17)∣∣∣Dn

ξDmx

(xαξβql (x,ξ)

)∣∣∣≤m

∑j=0

n

∑k=0

(mj

)(nk

)Cα, jCβ,k

∣∣∣xα− jξβ−kDn−kξ Dm− j

x ql (x,ξ)∣∣∣

Using (16), we obtain

(18)∣∣∣xα− jξβ−kDn−k

ξ Dm− jx ql (x,ξ)

∣∣∣=∣∣∣xkγ∣∣∣∣∣∣xα− j−kγξβ−k

∣∣∣ 1+ |ξ|k

1+ |ξ|k·1+ |x| j

1+ |x| j

·∣∣∣Dn−k


∣∣∣

≤∣∣∣xα− j−kγξβ−k

∣∣∣(

1+ |ξ|k)(

1+ |x| j)〈x〉kγ〈ξ〉−k〈x〉− j

· |q0(x,ξ)| 〈ξ〉−(n−k+l)〈x〉−(m− j+l)+γ(n−k+l).

Now, it can easily be seen that

(19)∣∣∣xα− j−kγξβ−k

∣∣∣(

1+ |ξ|k)(

1+ |x| j)≤C|p(x,ξ)| ,

in fact every term on the first member of (19) has exponent thatbelongs to the Newton’spolyhedronN p of the polynomialp(x,ξ).

Using (17), (18), (19), we obtain∣∣∣xα− jξβ−kDn−k


∣∣∣≤Cn,k,m, j 〈ξ〉−(n+l)〈x〉−(m+l)+γ(n+l),

soxαξβql (x,ξ) ∈ SG(−l ,−l+γl)(1,1),(0,γ) , l = 0,1,2, . . . .

We denoter(x,D) = xαDβQ. By Theorem 3, the symbolr(x,ξ) has the follow-ing asymptotic expansion:

(20) r(x,ξ) ∼ xαξβq(x,ξ)+

(β1

)xαξβ−1Dxq(x,ξ)

+ . . .+

(β

β−1

)xαξDβ−1

x q(x,ξ)+xαDβxq(x,ξ).

We note that the sum in (20) is a finite sum. By construction,q(x,ξ) has asymptotic

expansionq(x,ξ) ∼∞

∑l=0

ql (x,ξ) where theql (x,ξ) are defined in (12),l = 0,1,2, . . . .

It is obvious that

xαξβq(x,ξ) ∼∞

∑l=0

xαξβql (x,ξ).


For everyxαξβql(x,ξ), l = 0,1,2, . . . , the hypothesis of the Theorem 2 are satisfiedwith m0 = (0,0), therefore it exists a symbols(x,ξ) ∈ SG0,0

ρ,δ that has the asymptotic

expansion∞

∑l=0

xαξβql (x,ξ). We conclude that symbolss(x,ξ) andxαξβq(x,ξ) have the

same asymptotic expansion, then the symbolxαξβq(x,ξ) belongs toSG0,0ρ,δ.

Using the properties of the Newton’s polyhedron associatedto the symbol (13)and arguing similarly, we prove that

xαξβ− jD jxq(x,ξ) ∈ SG(− j ,− j+γ j)

(1,1),(0,γ) , j = 1, . . . ,β.

So we have proved thatr(x,ξ) ∈ SG(0,0)(1,1),(0,γ). By Proposition 4, we have thatxαDβQ is

continuous fromL2 into itself.

LetN p be the Newton’s polyhedron of the symbol (13).

DEFINITION 6. We set

H∗ =

u∈ S ′/xαDβu∈ L2, (α,β) ∈ N p

,

with norm‖u‖H∗ = ∑

0≤β≤m10≤α≤γβ

∥∥∥xαDβu∥∥∥

L2.

THEOREM 7. Let P be the operator with symbol(13). The following inequalityholds for any t∈ R2, for a suitable C> 0:

‖u‖H∗ ≤C(‖Pu‖L2 +‖u‖Ht ) , u∈ L2.

Proof. The operatorP is hypoelliptic, therefore, if we callQ its parametrix, we have

(21) QPu= u+Ku,

whereK ∈ OPSG−∞. Obviously, we can write (21) asu = QPu−Ku. Using the normH∗ we have

‖u‖H∗ ≤ ‖QPu‖H∗ +‖Ku‖H∗ .

We would like to prove that‖QPu‖H∗ ≤C‖Pu‖L2. This is true if the operatorQ : L2 →

H∗ is continuous. By definition ofH∗ norm, this is equivalent to ∑0≤β≤m10≤α≤γβ

∥∥∥xαDβQv∥∥∥

L2≤

C‖v‖L2. In other words, we have to prove thatxαDβQ : L2 → L2 is continuous. UsingProposition 5 we know that this assertion is valid.

It is easy to prove that‖Ku‖H∗ ≤C‖u‖Ht for anyt ∈ R2.

Observe finally that the boundedness of the parametrixQ : L2 → H∗ implies theFredholm property ofP. Namely, we haveP : H∗ → L2, as evident from Definition 6.MoreoverPQ= I +K1, QP= I +K2, whereK1,K2 are regularizing, hence compact inH∗ andL2. It follows that the mapP : H∗ → L2 is Fredholm.

112 I. Camperi

References

[1] BEALS R.,A general calculus of pseudodifferential operators, Duke Math. J.42 1 (1975), 1–42.

[2] BEALS R. AND FEFFERMAN C.,Spatially inhomogeneous pseudodifferential operators, I, Comm. onPure and Applied Math.XXVII (1974), 1–24.

[3] BOGGIATTO P., BUZANO E., RODINO L., Global hypoellipticity and spectral theory, Akademie Ver-lag, Berlin 1996.

[4] CAPPIELLO M., GRAMCHEV T., RODINO L., Subexponential decay and uniform holomorphic exten-sions for semilinear differential equations, preprint, 2006.

[5] CORDESH. O.,The technique of pseudodifferential operators, University Press, Cambridge 1995.

[6] H ORMANDERL., The analysis of linear partial differential operators III, Springer-Verlag, Berlin 1985.

[7] GINDIKIN S. AND VOLEVICH L. R., The method of Newton’s polyhedron in the theory of partialdifferential equations, Mathematics and its applications (Soviet Series) 86, 1992.

[8] K UMANO-GO H., Pseudo-differential operators, MIT Press, Cambridge 1981.

[9] PARENTI C., Operatori pseudodifferenziali inRn e applicazioni, Ann. Mat. Pura Appl.93 (1972),359–389.

[10] SHUBIN M. A., Pseudo-differential operators and spectral theory, Springer-Verlag, Berlin 1987.

AMS Subject Classification: 35H10, 35S05.

Igor CAMPERI, Dipartimento di Matematica, Universita degli Studi di Torino, Via Carlo Alberto, 10,10123, Torino, ITALIAe-mail: [email protected]

Lavoro pervenuto in redazione il 15.02.2007.


M. Lisi – S. Totaro

ANALYSIS OF AN AGE-STRUCTURED MSEIR MODEL

Abstract. In this paper we study the properties of an age-structured MSEIR epidemic model.Existence, uniqueness and positivity of the strict solution of the problem are proved by usingsemigroup techniques. Moreover, the expression of the total population density by means ofa suitable nonlinear semigroup is given. Finally, a semplified system of ODE is proposed andthe equilibrium solutions with stability results are analized.

1. Introduction

In the framework of epidemic models, many kinds of mathematical models have beeninvestigated: the most common ones are the so called SIR and SIS models. The firsttype analyzes the theoretical number of people infected by acontagious illness. In themodeling transmission dynamics of a communicable disease,it is common to dividethe population into distint classes whose sizes change withtime. The name SIR isdue to the fact that this kind of models involves ordinary differential equations for thenumber of Susceptible, Infected and Recovered individuals: one the simplest examplesis the Kermack-McKendrick model, proposed to study London’s cholera of 1865 andBombay’s plague of 1906, [6]. Talking about the second type,SIS models take intoconsideration a group of diseases for which infection does not confer immunity (e.g.,gonorreha); a Susceptible can become Infective and Susceptible again, [8].

In most of cases, the set of individuals studied in SIR and SISmodels dependsonly on timet, whereas the variable agea does not effect the epidemic dynamics.However, since many diseases, such as measles or chickenpox, are primarily diseasesof children (see, for istance, [2], [3] and [4]), only subdividing the population intodiffering age-classes we can be able to capture age-structured transmission in moredetail.

In this work, we shall study an age-structured MSEIR epidemic model, where,besides Susceptibles (S(a,t)), Infectious (I(a,t)) and Removed (R(a,t)), a class of indi-viduals protected by Maternal antibodies after their birth(M(a,t)) and a class of indi-viduals Exposed to the disease but not yet infected (E(a,t))are considered. This lastchoice is due to the fact that many diseases, such as measles and mumps, are charac-terized by a latent period. Literature in the framework of MSEIR models is not wide:an example is given by [13] (see also the references quoted therein), where existenceand uniqueness of positive steady states for an age-structured MSEIR epidemic model,similar to that proposed here, is proved. Our model is improved considering the totalpopulationN not only depending on the age variablea, but also on the time variablet.Moreover, whereas in [13] many manipulations to the model are made and the authorsused the positive invariant sets theory, we study the problem simply as a semilinearproblem, by using the classical semigroup theory, ([1],[9]). Moreover, we prove thatthe nonlinear term of the system is Frechet differentiable and the expression of the total

113

114 M. Lisi – S. Totaro

population density, by means of a suitable nonlinear semigroup, is given.

The paper is organized as follows: after the epidemic model description, inSection 2, by using semigroup techniques ([1],[9]), we studies global existence anduniqueness of the model solution and we prove that M, S, E, I, R(i.e., the densitiesof individuals protected by maternal antibodies, susceptibles, exposed, infective andremoved, respectively) are positive and regular functions. The results provide the ex-plicit form of the total population density and also a prioriestimates of the functionsS, E, I, which allow to get positivity and global existence results. Finally, under somehomogeneity assumptions, last Section is devoted to derivea system of ordinary dif-ferential equations for the total number of individuals protected by maternal antibody,susceptibles, latents, infectives and removed ones; the equilibrium solutions and theirstability are also analyzed. Note that the results of this paper agree with those presentin literature (see, for instance, [11]).

A further work will be dedicated to the study of an MSEIR modelwith vacci-nation effects: the idea is that of using mathematical techniques similar to those usedin the present paper (see [12], for numerical simulation).

2. The epidemic model

Assume an isolated population of individuals of age a, at time t (no immigration oremigration process is considered), can be divided into a setof five disjoint classes,dependent upon their experience with respect to a given disease. Related to each groupwe consider:

1. M = M(a, t): density of individuals protected by maternal antibodies;

2. S= S(a, t): density of susceptibles;

3. E = E(a, t): density of people exposed to the disease but not yet infectious (la-tents);

4. I = I(a, t): density of infectives;

5. R= R(a, t): density of removed (or immunes).

The total population densityN = N(a,t) will be such that:

(1) N = M +S+E+ I +R, 0≤ a≤ rm,t ≥ 0,

with rm < +∞ the highest age attended.

Analysis of a MSEIR model 115

The spread of disease can be described by the following system of partial integro-differential equations:

(2)

Mt +Ma = −(µ+ δ)M,

St +Sa = δM− (µ+ λ)S,

Et +Ea = λS− (µ+ α)E,

It + Ia = αE− (µ+ γ)I ,

Rt +Ra = γI −µR,

where

Ht =∂∂t

H(a, t), Ha =∂

∂aH(a,t), H = M,S,E, I ,R, a∈ (0, rm),t > 0,

the costantsδ−1,α−1,γ−1 represent the mean period protected by maternal antibodies,the mean latent period and the mean infectious period, respectively, whereas withµ =µ(a) we indicate the instantaneous death rate at agea. In order to have attainable finiteage, we assume that:

Hp1: µ is a nonnegative locally integrable function, such thatR rm

0 µ(a)da= +∞.

Finally, the termλ = λ(a, t) represents the so-called force of infection and it is givenby the following function:

λ(a, t) =

Z rm

0β(a,σ)I(σ,t)dσ,

whereβ(a,σ) is the probability, per unit of time, that a susceptible of age a meets aninfectious of ageσ and the first becomes latent: this means that, at timet, the probabil-ity a susceptible became latent during the interval(a,a+ da) is given byλ(a,t)da. Itis reasonable to assume:

Hp2: β(a,σ) is an essentially bounded function over the interval(0, rm)× (0, rm):0≤ β ≤ β.

System (2) is supplemented with the boundary conditions:

M(0, t) = q > 0, H(0,t) = 0, H = S,E, I ,R

and the initial conditions:

H(a,0) = H0(a) > 0, H = M,S,E, I ,R.

REMARK 1. From now on, we often use the following definition:

H = Hl(a), H = N,M,S,E, I ,R,


with

l(a) = exp[Z a

0µ(s)ds].

and the following notations:

Ht =∂∂t

H(a, t), Ha =∂

∂aH(a,t), H = N,M, S, E, I ,R, a∈ (0, rm),t > 0,

H0(a) = H0(a)l(a), H = N,M, S, E, I ,R, a∈ [0, rm].

By summing all equations of system (2) and taking into account definition (1),we have the following system:

(3)

Nt + Na = 0,

N(0, t) = q,

N(a,0) = N0(a)l(a) = N0(a),

whereN0(a) = M0(a)+S0(a)+E0(a)+ I0(a)+R0(a) (see also Remark 1).

Let us consider the Banach spaceX = L1(0, rm), with norm

(4) ‖ f‖ =

Z rm

0| f (a)|da, ∀ f ∈ X,

and positive cone ([7], [10])

X+ = f ∈ X, f (a) ≥ 0,a.e. in(0, rm).

It is easy to prove that the solution of system (3) has the form:

(5) N(a, t) =

N0(a− t), a≥ t,q, a < t.

If N0 is a bounded function, a suitablek > 0 exists such that|N0(a)| ≤ k, for anya∈ [0, rm] and

(6) |N(a, t)| ≤ K, ∀t ≥ 0,a∈ [0, rm],

with K = max(k,q).

Since the first equation of system (2) is similar to the first one of (3), the solutionof

Mt + Ma = −δM,

M(0, t) = q,

M(a,0) = M0(a)l(a) = M0(a),

is given by:

(7) M(a, t) =

M0(a− t)exp(−δt))], a≥ t,qexp(−δa), a < t.


REMARK 2. Note thatM ∈ X+, provided thatM0(a) ≥ 0 for a∈ [0, rm]. More-over, since it is reasonable to assume|M0(a)| ≤ k, thenM satisfies estimate (6).

Since from definitions (1) and Remark 1

R= N− M− S− E− I ,

system (2) will be completely solved, if we find the solutionsof the following system :

(8)

St + Sa = δM−λS,

Et + Ea = λS,

It + IaαE = γIS(0, t) = E(0, t) = I(0,t) = 0,

S(a,0) = S0(a), E(a,0) = E0(a), I(a,0) = I0(a),

whereM is given by (7) (see also Remark 1).

Define the operator:

(9) L f = − f ′, D(L) = f ∈ X, f ′ ∈ X, f (0) = 0, R(L) ⊂ X.

Note thatL is a linear operator, because its domain contains the homogeneous boundarycondition. We prove it generates a linearC0−semigroup, [1].

In order to study system (8), we prove three preliminary lemmas.

LEMMA 1. The operator L satisfies the following properties:

i) ‖(λI −L)−1g‖ ≤1λ‖g‖, ∀g∈ X;

ii) D(L) is dense in X;

iii) L is closed.

Proof. Consider the equation(λI −L) f = g

where the unknownf must be sought inD(L). Hence the solution of

f ′ = −λ f +g, f (0) = 0,

is given by:

f (a) =

Z a

0g(s)exp[−λ(a−s)]ds, 0 < a < rm.

Moreover, from (4):

‖ f‖ ≤Z rm

0eλs|g(s)|ds

Z rm

se−λada≤

1λ‖g‖.

Thus, f belongs toD(L). Propertyii) follows from the fact thatD(L) ⊃ C∞0 (0, rm),

which is dense inX. Finally, sinceD((λI − L)−1) = X and(λI − L)−1 is a boundedoperator,(λI −L)−1 is a closed operator. Hence, alsoL = −(λI −L)+ λI is a closedoperator, [5].


REMARK 3. Lemma 1 proves that the operator(λI −L)−1 exists, it is definedon the whole spaceX and is a bounded operator; hence, it is the resolvent operatorR(λ,L) of L, [5].

LEMMA 2. L is the generator of a C0−semigroup Z(t) = exp(tL),t ≥ 0, suchthat

‖Z(t) f‖ ≤ ‖ f‖.

The semigroup Z(t) maps the positive cone X+ into itself.

The proof of the theorem follows directly from the Hille Yosida Theorem, [5],[9].

Define the operator

J f =

Z rm

0β(a,σ) f (σ)dσ, D(J) = X, R(J) ⊂ X.

The following lemma holds.

LEMMA 3. J is a bounded operator and‖J‖ ≤ βrm. Moreover, J f∈ L∞(0, rm),for any f ∈ X, and‖J f‖∞ ≤ β‖ f‖, where‖ · ‖∞ is the norm in L∞(0, rm).

To study system (8), consider the Banach space

X∗ = X×X×X,

with norm

‖f‖∗ = ‖

f1f2f3

‖ =

3

∑i=1

‖ fi‖, ∀f ∈ X∗,

whereX = L1(0, rm) and‖ fi‖ is given by (4), withfi ∈ X, i = 1,2,3.

Define the following operators

L∗f =

L f1 0 00 L f2−α f2 00 0 L f3− γ f3

,

D(L∗) = D(L)×D(L)×D(L), R(L∗) ⊂ X∗,

(10) F f =

− f1J f3f1J f3α f2

, D(F) = X∗, R(F) ⊂ X∗.

By using operatorL∗ andF, the abstract version of (8) becomes:

(11)

ddt

u(t) = L∗u(t)+Fu(t)+g(t), t > 0,

u(0) = u0 ,


where

(12) u(t) =

u1(t)u2(t)u3(t)

=

S(.,t)E(.,t)I(.,t)

; g(t) =

δM(.,t)00

; u0 =

S0

E0

I0

.

LEMMA 4. The linear operator L∗ is the generator of a C0-semigroupexp(tL∗), t ≥ 0, given by

exp(tL∗) =

exp(tL) 0 00 exp(−αt)exp(tL) 00 0 exp(−γt)exp(tL)

,

with L given by (9). Moreover,

‖exp(tL∗)‖ ≤ 1.

Let r be a given positive number, the following subset ofX∗ can be defined

Dr = f ∈ X∗,‖f‖∗ ≤ r.

LEMMA 5. F satisfies a Lipschitz condition on Dr and

(13) ‖F(f)−F(g)‖∗ ≤ l‖f −g‖∗,

with l = 2βr + α.

Proof.

‖F(f)−F(g)‖∗ = ‖ f1J f3−g1Jg3‖+‖ f1J f3−g1Jg3‖+‖α f2−αg2‖ ≤

≤ 2‖ f1J f3− f1Jg3‖+2‖ f1Jg3−g1Jg3‖+ α‖ f2−g2‖, ∀f,g∈ Dr .

From Lemma 3, we have:

‖F(f)−F(g)‖∗ ≤ 2βr‖ f3−g3‖+2βr‖ f1−g1‖+ α‖ f2−g2‖ ≤ (2βr + α)‖f −g‖∗.

LEMMA 6. F is Frechet differentiable in X∗ and the derivative Ff is continuouswith respect tof ∈ X∗.

Proof. The following equality must be proved, for any givenf,h ∈ X∗:

F(f +h)−F(f) = Ff (h)+G(f,h) ,

whereFf is a linear bounded operator which depends onf ∈ X∗ andG(f,h) is such that

lim‖h‖∗→0

G(f,h)

‖h‖∗= 0.


Since

F(f +h)−F(f) =

− f1Jh3−h1J f3−h1Jh3

f1Jh3 +h1J f3 +h1Jh3

αh2

,

by defining

Ff(h) =

− f1Jh3−h1J f3f1Jh3+h1J f3

αh2

, G(f,h) =

−h1Jh3

h1Jh3

0

,

we have that

lim‖h‖∗→0

‖G(f,h)‖∗‖h‖∗

= lim‖h‖∗→0

2β‖h‖∗ = 0.

Since‖Ff(h)‖∗ = 2‖ f1Jh3+h1J f3‖1 + α‖h2‖1 ≤

(4β‖f‖∗+ α

)‖h‖∗,

Ff is linear and bounded. Moreover, since

‖ f1Jh3 +h1J f3−g1Jh3−h1Jg3‖ ≤ ‖( f1−g1)Jh3‖+‖h1(J f3−Jg3)‖ ≤

≤ β‖h3‖‖( f1−g1)‖+ β‖h1‖‖ f3−g3‖ ≤ 2β‖h‖∗‖f −g‖∗,

the continuity ofFf with respect tof ∈ X∗ is proved, i.e.,

lim‖f−g‖∗→0

‖Ff(h)−Fg(h)‖∗ = 0.

THEOREM 1. System (11) has a unique strict solutionu = u(t) defined on asuitable interval[0,T].

The proof of the theorem follows directly from Lemmas 4, 5, 6 and by thedefinition ofg, given by (12), [1]. Such a solution can be found by using a successiveapproximation procedure for the integral equation

(14) u(t) = exp(tL∗)u0 +

Z t

0exp[(t −s)L∗] [g(s)+F(u(s))]ds.

Since we proved thatN(a, t) is a bounded and positive function (see (6)), it is quitereasonable to think that also the solution of (11) (or equivalently the solution of theintegral equation (14)) has bounded and positive components. In order to prove this,let us introduce the space

X∗∞ = X∞ ×X∞×X∞ = L∞(0, rm)×L∞(0, rm)×L∞(0, rm) ,

with norm

‖f‖∞ =3

∑i=1

‖ fi‖∞, ∀f ∈ X∗∞,


where‖ fi‖∞ = sup fi(a),a∈ (0, rm),∀ fi ∈ X∞, i = 1,2,3. For a suitable ˆm> 0, definethe set:

S(m) = f ∈ X∗∩X∗∞,‖f‖∞ ≤ m.

It can be proved thatS(m) is a not empty closed set ofX∗ (see Example 1.23 of [1]).

LEMMA 7. If f ∈ S(m), thenexp(tL∗)f ∈ S(m) and F(f) ∈ S(2βm2 + αm).

The proof of the lemma follows from Lemmas 3, 4, definition (10) and theproperties ofS(m).

Define the spaceX∗

c = C([0,T];X∗) ,

(T will be chosen in the sequel), with the norm

‖f‖c = max‖f(t)‖∗,t ∈ [0,T], ∀f ∈ X∗c .

Consider the following closed subset ofX∗c :

∆(m) = f ∈ X∗c , f(t) ∈ S(m),t ∈ [0,T].

The nonlinear Volterra integral equation (14) can be written as

u = Qu,

where:

(15) Q(f(t)) = exp(tL∗)u0 +

Z t

0exp[(t −s)L∗] [g(s)+F(f(s))]ds,

with D(Q) = X∗c ,R(Q) ⊂ X∗

c .

LEMMA 8. If u0 ∈ S(n), with n a given positive constant, then:

i) Qf ∈ ∆(m p(T)), for f ∈ ∆(m);

ii) ‖Q(f)−Q(g)‖c ≤ p(T)‖f −g‖c, for f,g∈ ∆(m),

where

p(T) =

[n+expT −1

m

(δK +2m2β+2m2βrm+ αm

)].

Proof. From definitions (15), we have:

‖Q(f(t))‖∞ ≤ n+(δK +2βm2 + αm)

Z t

0exp(t −s)ds≤ mp(T) .

Moreover, iff,g∈ ∆(m), from Lemma 5, we have:

‖Q(f)−Q(g)‖c ≤

Z t

0exp(t −s)‖F(f(s))−F(g(s))‖c ds≤ p(T)‖f −g‖c.


REMARK 4. If we choose ˆn < m, then limT→0

p(T) = n/m < 1, i.e., a suitably

small T exists, such thatp(T) < 1. This means thatQ maps∆(m) into itself and isstrictly contractive on∆(m). Note that the relevance of this result is due to the fact that,even if T could be really small, after finding an a priori estimate of the norm of thesolution, it permits to prove the existence and uniqueness of the solution for eacht ≥ 0(see in the sequel and [9], Theorem 1.4, Chapter 6).

THEOREM 2. If u0 ∈ S(n) and m > n, then system (11) has a unique strictsolutionu = u(t) defined on a suitable interval[0,T] and such thatu(t) ∈ S(m),t ∈[0,T].

Chooseu0 =

S0

E0

I0

∈ S(n) andm> n. From system (11), we have:

u1(t) = exp(tL)e−mtS0 +

Z t

0exp[(t −s)L]e−m(t−s)δM(s)ds+

+

Z t

0exp[(t −s)L]e−m(t−s)u1(s)(m−Ju3(s))ds,

(16) u2(t) = exp(tL)e−αt I0 +

Z t

0exp[(t −s)L]e−α(t−s)u1(s)Ju3(s)ds,

u3(t) = exp(tL)e−γt E0 +Z t

0exp[(t −s)L]e−γ(t−s)αu2(s)ds.

THEOREM 3. Each component ui(t), i = 1,2,3, of u(t) belongs to X+, for anyt ui is defined.

It is possible to prove the theorem by a successive approximation procedure.

From Theorem 3, since it can be easily proved thatM,R∈ X+, from (6) wehave that‖u(t)‖∗ ≤ Krm. This a priori estimate proves the existence of a unique strictsolutionu(t) of the Eq. (14), for eacht ≥ 0 (see [1] and [9], Theorem 1.4, Chapter6)). From Lemma 6 and the regularity properties of the known termg(t), follows thedifferentiability ofu(t). It is easy to prove that alsoM andRare differentiable functionsboth with respect toa andt. Hence, the following result holds (see [9], Theorem 1.5,Chapter 6).

THEOREM 4. System (8) has a unique classical solution defined for each t≥ 0and whose components belong to X+, provided that the initial conditions belong to X+.

REMARK 5. Defining the nonlinear operator

A f = − f ′, D(A) = f ∈ X, f ′ ∈ X, f (0) = q,


it can be easily proved thatA generates a nonlinear semigroup of contractionsW(t),t ≥0 given by (5) and the solution of the evolution problem is given byw(t)=W(t)N0,t ≥0. Note that the operatorA is nonlinear because of its domain.

3. A simplified model

The aim of this section is to derive a system of ordinary differential equations whichsimplify system (2) and to study the stability properties ofits equilibrium solutions. Inorder to make this section more readable and less boring, we introduce some notationsand omit to write many calculations.

Define the following quantities:

(17) µH =

R rm0 µ(a)H(a,t)da

‖H(t)‖, H = M,S,E, I ,R,

m=

R rm0 p(a)S(a, t)da

‖S(t)‖, with p(a) =

R rm0 β(a,σ)I(σ,t)dσ

‖I(t)‖, a∈ (0, rm).

To get a simplified model, we make the following assumption:

Hp3: the quantities defined in (17) are constant and such that:

µ= µM = µS = µE = µI = µR.

Even if Hp3 seems very restrictive, estimates found in Section 2 can be used to evaluateµH ,H = M,S,E, I ,R.

Integrating equations of system (2) with respect toa in the interval[0, rm], takinginto account the properties ofM,S,E, I ,R, and using the notation‖H‖t = d

dt‖H‖,H =M,S,E, I ,R, we have:

(18)

‖M(t)‖t = q− (µ+ δ)‖M(t)‖ ,

‖S(t)‖t = δ‖M(t)‖− µ‖S(t)‖−M ‖S(t)‖‖I(t)‖ ,

‖E(t)‖t = m‖S(t)‖‖I(t)‖− µ‖E(t)‖−α‖E(t)‖ ,

‖I(t)‖t = α‖E(t)‖− µ‖I(t)‖− γ‖I(t)‖ ,

‖R(t)‖t = γ‖I(t)‖− µ‖R(t)‖ ,

‖H(0)‖ = ‖H0‖ H = M,S,E, I ,R.

Solving the first equation of (18), we get

(19) ‖M(t)‖ = e−(µ+δ)t ‖M0‖+q

µ+ δ

(1−e−(µ+δ)t

).


Note thatlim

t→+∞‖M(t)‖ =

qµ+ δ

= M,

whereM is the equilibrium solution and it results asymptotically stable.

By summing all equations of system (18), and using (19), it iseasy to prove that

limt→+∞

(‖S(t)‖+‖E(t)‖+‖I(t)‖+‖R(t)‖) =qµ−

qµ+ δ

=qδ

µ(µ+ δ).

In order to simplify (18), we shall consider the system for large time, and we put‖M(t)‖ = M:

(20)

‖S(t)‖t =δq

µ+ δ− (µ+m‖I(t)‖)‖S(t)‖ ,

‖E(t)‖t = m‖S(t)‖‖I(t)‖− (µ+ α)‖E(t)‖ ,

‖I(t)‖t = α‖E(t)‖− (µ+ γ)‖I(t)‖ ,

‖R(t)‖t = γ‖I(t)‖− µ‖R(t)‖ ,

‖H(0)‖ = ‖H0‖ H = S,E, I ,R.

The equilibrium solution(S,E, I ,R) of (20) satisfies the following system:

(21)

0 =δq

µ+ δ− (µ+mI)S,

0 = mSI − (µ+ α)E,

0 = αE− (µ+ γ) I ,

0 = γI − µR.

From the last two equations of (21), we get

(22) R=γµ

I , E =µ+ γ

αI .

Hence, by substituting (22) into the first two equations of (21), we have

(23)

0 =δq

µ+ δ− µS−mSI ,

0 = mSI −(µ+ α)(µ+ γ)

αI .

I = 0 is a solution of the second equation of (23); since from (22), we getR= E = 0,from the first of (23):

(24) S=δq

µ(µ+ δ).


If we define:

(25) S (t) = ‖S(t)‖−S, E (t) = ‖E(t)‖, I (t) = ‖I(t)‖,

system (20) becomes:

(26)

S t = −(µ+mI )S −mδq

µ(µ+ δ)I ,

E t = [mS +mδq

µ(µ+ δ)]I − (µ+ α)E ,

It = αE − (µ+ γ)I ,

S (0) = S0,E (0) = E0, I (0) = I0.

Since the Jacobian matrix of the right hand side of (26), evaluated in the equilibriumsolution(0,0,0) is

J(0,0,0) =

−µ 0 −mδq

µ(µ+ δ)

0 −(µ+ α)mδq

µ(µ+ δ)

0 α −(µ+ γ)

,

its eigenvalues are given by:

λ1 = −µ, λ2,3 =−(µ+ α)− (µ+ γ)±

√(α− γ)2 +4αmδq/µ(µ+ δ)

2.

Whereasλ1 is always a negative real number,λ2 6= λ3 are negative real numbers if

(µ+ α)(µ+ γ) >αmδq

µ(µ+ δ).

With this assumption, the equilibrium solution(0,0,0) is asymptotically stable; thenumber of susceptibles‖S(t)‖ tends toS, whereas both the number of latents and thenumber of infectives tend to zero. Moreover, it can be shown that also the number ofremoved individuals tends to zero, for large time. Note that, if

(27) (µ+ α)(µ+ γ) =αmδq

µ(µ+ δ),

one of the eigenvalues ofJ(0,0,0) is zero; hence the stability of(0,0,0) has to beanalized in another way. For example, by summing all equations of (26) with thefourth of (20) and definingR (t) = ‖R(t)‖, since

S (t)+E (t)+ I (t)+R (t) = (S0 +E0+ I0 +R0)e−µt,


we get

(28) limt→+∞S (t)+E (t)+ I (t)+R (t) = 0.

If S0,E0, I0,R0 ∈ R+, (‖S0‖ > S from (25)), henceS (t),E (t), I (t),R (t) ∈ R+, t ≥ 0and (28) implies that each term of the sum tends to 0. In particular, if ‖S0‖ > S (forinstance,δ or q or both are small quantities), the number of susceptibles decreases toS, whereas the number of latent, infectious and removed individuals tends to zero.

In a similar way, ifS0 < 0 (‖S0‖< Sfrom (25)),E0, I0,R0 ∈R+, henceS (t) < 0,E (t), I (t),R (t) ∈ R+, t ≥ 0. In fact, with (27), system (26) becomes:

S t = −(µ+mI )S − [(µ+ α)(µ+ γ)I ]/α,

E t = m[S +(µ+ α)(µ+ γ)/α]I − (µ+ α)E ,

It = αE − (µ+ γ)I ,

R t = γI − µR ,

S (0) = S0,E (0) = E0, I (0) = I0,R (0) = R0.

Since by summing the second, the third and the fourth equations of the previous system

(E + I +R )t = m[S +(µ+ α)(µ+ γ)]I /α− µ(R +E + I ),E (0)+ I (0)+R (0) = E0 + I0 +R0,

by applying Gromwall’s inequality, we get:

E (t)+ I (t)+R (t) ≤ (E0 + I0+R0)exp[(mS− µ)t],

where we used (24) and (27). As a consequence, if

(29) S<µm

,

the sumE (t)+ I (t)+R (t) goes to zero, ast →+∞. SinceE , I ,R ∈R+, it results thatlim

t→+∞E (t) = lim

t→+∞I (t) = lim

t+→∞R (t) = 0. Hence, from (28), we get that lim

t→+∞S (t) = 0,

that is limt→+∞

‖S(t)‖ = S. Thus, if (27) holds and (29) is fulfilled (for instance, if the

latent period or the force of infection or both are small), the equilibrium(S,0,0,0) issuch thatStends toS.

Finally, if

(30) (µ+ α)(µ+ γ) <αmδq

µ(µ+ δ),

one of the eigenvalues ofJ(0,0,0) is positive and the equilibrium solution(0,0,0) isunstable.


Let us come back to system (23); ifI 6= 0 the solutions are given by

(31) S=(µ+ α)(µ+ γ)

mα, I = h, E = h

(µ+ γ

α

), R= h

γµ,

where

(32) h =δqα

(µ+ δ)(µ+ α)(µ+ γ)−

µm

.

Values in (31) have biological meaning ifh > 0. Since this agrees with (30), we havethat the condition which makes the equilibrium solution(S,0,0,0) unstable, providesalso the existence of the biological equilibrium solution (S ,E , I ,R ). Moreover,h = 0represents (27) and yields to the equilibrium solution(S,0,0,0) again.

Assumingh > 0, let us analyze the stability of(S, E, I ,R). Since, by usingdefinitions (31) and the following notation

H (t) = ‖H(t)‖− H, H = S, I ,R,

system (20) becomes:

(33)

S t = −m‖S‖I − (mh− µ)S ,

E t = m(‖S‖I +hS )− (µ+ α)E ,

It = αE − (µ+ γ) I ,

the unique equilibrium solution is(0,0,0) and the Jacobian matrix of the right handside is:

J(0,0,0) =

−(mh+ µ) 0 −mS

mh −(µ+ α) mS

0 α −(µ+ γ)

.

Since the characteristic equation ofJ(0,0,0)

(34) λ3 + λ2(a+b+h+ µ)−λ(a+b)(h+ µ)+abh= 0,

(with a = µ+ α,b = µ+ γ) has three solutions whose product is given by the knowntermabh> 0, it follows that the matrixJ(0,0,0) has at least one positive eigenvalue.In fact, if (34) admits only one real solution, it must be positive, because the product ofthe other two complex conjugate solutions is positive. On the other hand, if there arethree real solutions, by examining all the possibilities, we find that at least one of theseis positive. This means that the equilibrium point(0,0,0) is unstable for system (33)and thus the equilibrium solution(S, E, I ,R) is unstable for system (20).

Note that the stability analysis is quite similar to that made for other epidemicmodels (see, for instance, the more complicated model presented in [11]). In particular,


we can conclude that if the coefficientm is very small or the mortality coefficientµ isvery high, the number of susceptibles tends to the equilibrium valueS 6= 0, whereasthe number of latent, infective and removed individuals tends to zero. In this case, asin the simpler Kermack Mc Kendrick model, the disease stops because there are notyet infectives and there are some individuals who do not become infective in any case.On the other hand, ifh > 0 (see (32)), i.e., if the mortality coefficientµ is small orm is high, there is a non trivial equilibrium solution, (endemic steady state solution)(S, E, I ,R), which is unstable (see [11]).

All the results of this section can be summarized in the following theorem.

THEOREM5. The simplified system (20) has two equilibrium solutions(S,0,0,0)and (S, E, I ,R). The first solution is asymptotically stable if h> 0 and is unstable ifh < 0, with h given by (32). The second solution has biological meaning if h> 0 and itis unstable. If h= 0, only the equilibrium solution(S,0,0,0) exists and it is such thatE , I ,R → 0 andS (t) → S, provided thatS< µ/m.

AcknowledgmentsThis work was partially supported by PAR 2006 - Research Project“Metodi e modelli matematici per le applicazioni” of the University of Siena, Italy, aswell as by M.U.R.S.T. research funds.

References

[1] BELLENI MORANTE A. AND MCBRIDE A. C., Applied Nonlinear Semigroups, John Wiley & Sons,Chichester 1999.

[2] BUSEMBERGS., IANNELLI M. AND THIEME H. R.,Global behaviour of an age-structured epidemicmodel, SIAM J. Math. Anal.22 (1991), 1065–1080.

[3] CHA Y., IANNELLI M. AND M ILNER F. A., Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model, Math. Biosci.150(1998), 177–190.

[4] EL-DOMA M., Analysis of an age-dependent SIS epidemic model with vertical transmission and pro-portionate mixing assumption, Math. Comput. Modelling29 (1999), 31–43.

[5] K ATO T., Perturbation theory for linear operators, Springer Verlag, 1984.

[6] K ERMACK W.O. AND MCKENDRICK A. G.,A contribution to the mathematical theory of epidemics,Proc. Roy. Soc. Lod. A115(1927), 700–721.

[7] K RASNOSEL’ SKII M. Positive solutions of operator equations, P. NoordHoof, Groningen 1964.

[8] M ANFREDI P. AND WILLIAMS J. R.,Realistic population dynamics in epidemiological models:theimpact of population decline on the dynamics of childwood infectious diseases. Measels in Italy as anexample, Math. Biosci.192(2004), 153–175.

[9] PAZY A., Perturbation theory for linear operators, Springer Verlag, 1984.

[10] SCHAEFFERH. H.,Banach lattices and positive operators, Grund. Math. Wissenschafeten, Band 215,Springer Verlag, New York 1974.

[11] SHUETTE M. C., A qualitative analysis for the transmission of varicella-zoster virus, Math. Biosci.182(2003), 113–126.

[12] SHUETTE M. C. AND HETHCOTEH. W., Modeling the effects of varicella vaccination programs onthe incidence of chickenpox and shingles, Bull. Math. Biol. 61 (1999), 1031–1064.

[13] L I X., GUPUR G. AND ZHU G., Existence and uniqueness of endemic states for the age structuredMSEIR epidemic model, Acta Math. Appl. Sinica18 (2002), 441–454.


AMS Subject Classification: 47D06, 47H20, 92B05.

Meri LISI, Silvia TOTARO, Dipartimento di Scienze Matematiche ed Informatiche “Roberto Magari”,Universita degli Studi di Siena, Pian dei Mantellini 44, 53100, Siena, ITALYe-mail: [email protected], [email protected]



H.L. Vasudeva – Mandeep Singh∗

WEIGHTED POWER MEANS OF OPERATORS

Abstract. Let σ1,σ2, · · · ,σn be positive real numbers satisfyingn∑

i=1σi = 1 andA1,A2, · · · ,An

be positive operators. LetMσ,γ(A) = (n

∑i=1

σiAγi )

1/γ, γ > 0. It has been shown that limγ→0+

Mσ,γ

exists. Some known inequalities have also been generalized.

1. Introduction

K. V. Bhagwat and R. Subramanian [2] considered the validityof the well known in-equalities between power means of a set of positive real numbers, when the latter arereplaced by positive operators on a Hilbert space. They proved an analogue of thearithmetic-harmonic mean inequality for positive numbersin the case of positive op-erators. In what follows, we extend the above said inequality to weighted means. Theprocedure for extension of the arithmetic-harmonic mean inequality to weighted meanssuggested in [2], though standard is yet involved. It turns out that the method of proofemployed in [2] when suitably modified, gives a direct proof of weighted arithmetic-harmonic mean inequality.

We show that limγ→0+

Mσ,γ(A), whereMσ,γ(A) = (n

∑i=1

σiAγi )

1/γ, σi (i = 1,2, ...,n)

are non-negative numbers withn∑

i=1σi = 1 andAi (i = 1,2, ...,n) are positive operators,

exists and equalsMσ,0(A) = expn

∑i=1

σi logAi. The existence of this limit has been

proved in [4], however, the method of proof therein is ratherlong and tedious. Wegive a direct proof of the existence of the limit. The value ofthe limit reduces to theusual generalized geometric mean in the case of commuting operators. The surprisingfact in the case of Hilbert space operators is that the generalized geometric mean is notless than or equal to the weighted arithmetic mean. There is areference to generalizedgeometric mean in [5] as well.

Some other inequalities proved in [2] have also been extended to weightedmeans.

∗The authors would like to thank Professor A. L. Brown for useful discussions.

131

132 H.L. Vasudeva – Mandeep Singh

2. Weighted Power Means

ThroughoutH will denote a Hilbert space andA j , 1≤ j ≤ n, are bounded self adjointpositive definite linear operators fromH to H . Let σ j , 1 ≤ j ≤ n, be positive real

numbers such thatn∑j=1

σ j = 1 andγ 6= 0 be a real number. SetMσ,γ(A) = (n∑j=1

σ jAγj)

1/γ.

The following theorem holds.

THEOREM 1. With notations as in paragraph above, we have

limγ→0+

(n

∑j=1

σ jAγj)

1/γ = expn

∑j=1

σ j logA j.

Proof. Observe that

n

∑j=1

σ jAγj =

n

∑j=1

σ jexpγlogA j

= I + γn

∑j=1

σ j logA j +n

∑j=1

σ j

∞

∑k=2

(γlogA j)k

k!

= I + γn

∑j=1

σ j logA j +∞

∑k=2

γk

k!

n

∑j=1

σ j(logA j)k.

So,(

n

∑j=1

σ jAγj

)1/γ

=

(I + γ

n

∑j=1

σ j logA j +∞

∑k=2

γk−1

k!

n

∑j=1

σ j(logA j)k

)1/γ

= exp

1γ log(I + γ

n

∑j=1

σ j logA j +∞

∑k=2

γk−1

k!

n

∑j=1

σ j(logA j)k)

= exp

1γ (

∞

∑l=1

(−1)l−1

lγl

n

∑j=1

σ j logA j +∞

∑k=2

γk−1

k!

n

∑j=1

σ j(logA j)kl )

= exp

n

∑j=1

σ j logA j +∞

∑k=2

γk−1

k!

n

∑j=1

σ j(logA j)k

+γ∞

∑l=2

(−1)l−1

lγl−2

∞

∑k=1

γk−1

k!

n

∑j=1

σ j(logA j)kl

.

Notice that for 0< γ < 1,

||∞

∑k=2

γk−1

k!

n

∑j=1

σ j(logA j)k|| ≤ γ

∞

∑k=2

1k!

n

∑j=1

σ j ||logA j ||k

≤ γ∞

∑k=2

1k!

max1≤ j≤n

||logA j ||k

≤ γ(

exp max1≤ j≤n

||logA j ||− min1≤ j≤n

||logA j ||−1

)

Weighted Power Means of Operators 133

and

||γ∞

∑l=2

(−1)l−1

lγl−2

(∞

∑k=1

γk−1

k!

n

∑j=1

σ j(logA j)k

)l

||

≤∞

∑l=2

||∞

∑k=1

γk

k!

n

∑j=1

σ j(logA j)k||l

≤∞

∑l=2

(∞

∑k=1

γk

k!max

1≤ j≤n||(logA j)

k||

)l

≤∞

∑l=2

(exp(γ max

1≤ j≤n||(logA j ||−1)

)l

≤∞

∑l=0

(exp(γ max

1≤ j≤n||(logA j ||−1)

)l

−1−exp(γ max1≤ j≤n

||(logA j ||)−1

=

1

1−

(exp

(γ max

1≤ j≤n||logA j ||

)−1

)

−exp(γ max1≤ j≤n

||logA j ||.

Consequently,

(n

∑j=1

σ jAγj

)1/γ

tends to expn

∑j=1

σ j logA j asγ → 0+ .

REMARK 1. The generalized geometric mean, namely, expn

∑j=1

σ j logA j, is

not necessarily less than the weighted arithmetic mean. Indeed,

expn

∑j=1

σ j logA j ≤n

∑j=1

σ jA j

implies the mapx→ ex is operator convex, which as is well known, is false in general(see [3] Problem V.5.1, p.147 ).


We shall need the following lemma in the sequel. For a proof (see [3] p.114).

LEMMA 1. If B is strictly positive and A≥ B then A is strictly positive andB−1 ≥ A−1 > 0.

THEOREM 2. For a set of positive operatorsA1,A2, · · · ,An on a Hilbert

spaceH , a set of positive real numbersσ1,σ2, · · · ,σn satisfyingn

∑i=1

σi = 1 and q≥

p≥ 1, Mσ,q(A) ≥ Mσ,p(A), where A= (A1,A2, · · · ,An).

Proof. The case 1≤ p= q is trivial, so we prove only the result forp < q, i.e., pq−1 <1. SetAq

j = B j andα = pq−1.

(n

∑j=1

σ jAqj )

α = (n

∑j=1

σ jB j)α ≥

n

∑j=1

σ jBαj =

n

∑j=1

σ jApj ,

using the concavity of the mapA→ Aα, 0 < α < 1, ([3] Th.V.2.5 & V.2.10). If p ≥

1, then the mapA → A1/p is order preserving (see [3]) and since(n

∑j=1

σ jAqj )

pq−1≥

n

∑j=1

σ jApj , it follows that

(n

∑j=1

σ j Aqj )

1/q ≥ (n

∑j=1

σ jApj )

1/p.

COROLLARY 1. For positive operators Ai , i = 1,2, ...n, q≤ p≤ −1 andσi >

0, i = 1,2, ...,n withn∑

i=1σi = 1, we have

Mσ,q(A) ≤ Mσ,p(A)

where A= (A1,A2, ...,An).

Proof. ReplacingA j by A−1j in Theorem 2, we obtain

(n

∑j=1

σ jA−q1j )1/q1 ≥ (

n

∑j=1

σ jA−p1j )1/p1,

for q1 ≥ p1 ≥ 1. Since the right hand side of the above inequality is positive, usingLemma 1 and then replace−q1, −p1 by q, p respectively, we obtain the desired in-equality.

THEOREM3. For positive operators Ai , i = 1,2, ...,n and positive real numbers

σi , i = 1,2, ...,n withn∑

i=1σi = 1, the following inequality holds

Mσ,γ(A) ≤ Mσ,1(A) ≤ Mσ,2γ(A)

Weighted Power Means of Operators 135

where A= (A1,A2, ...,An) and1/2≤ γ ≤ 1.

Proof. We need only to prove the first inequality as the last inequality is trivial fromTheorem 2. Now, since 1/2 ≤ γ ≤ 1, i.e., 1≤ γ−1 ≤ 2 and the functionx → x1/γ isoperator convex (see [3] Ex.V.2.11) it follows that

(n

∑i=1

σiAγi )

1/γ ≤n

∑i=1

σiAi ,

i.e.,Mσ,γ(A) ≤ Mσ,1(A).

COROLLARY 2. For positive operators Ai , i = 1,2, ...,n, and positive real num-

bersσi > 0, i = 1,2, ...,n withn∑

i=1σi = 1, we have

Mσ,2γ(A) ≤ Mσ,−1(A) ≤ Mσ,γ(A)

for −1≤ γ ≤−1/2.

Proof. From Theorem 3, we have

(1) (n

∑i=1

σiAγ1i )1/γ1 ≤

n

∑i=1

σiAi ≤ (n

∑i=1

σiA2γ1i )1/2γ1,

for 1/2≤ γ1 ≤ 1.

ReplacingAi by A−1i in (1), using Lemma 1 and then replacingγ1 by −γ, we

obtain the desired result.

THEOREM 4. For positive operators Ai , i = 1,2, · · · ,n and positive real num-

bers.σi , i = 1,2, · · · ,n withn∑

i=1σi = 1 the following inequality holds

(n

∑i=1

σiA−γi

)−1/γ

≤

(n

∑i=1

σiAγi

)1/γ

,

for γ ≥ 1.

Proof. Since the mapx→ x−1 is operator convex, it follows that(

n

∑i=1

σiA−1i

)−1

≤n

∑i=1

σiAi .

Now replacingAi by Aγi and using the order perserving property of the mapx→ xα, 0<

α ≤ 1 , we obtain

(n

∑i=1

σiA−γi )−1/γ ≤ (

n

∑i=1

σiAγi )

1/γ.


Finally, we have the following theorem (c.f. [1]).

THEOREM 5. Let q, p be real numbers. such that one of the following holds

(a) q≥ p q /∈ (−1,1), p /∈ (−1,1);(b) q≥ 1≥ p≥ 1/2; or(c) p≤−1≤ q≤−1/2.

ThenMσ,q(A) ≥ Mσ,p(A).

Proof. (a). If q≥ p≥ 1, the result follows from Theorem 2. If−1≥ q≥ p, the resultfollows from corollary 1. In case 1≤−p≤ q, we obtain from Theorem 2, Theorem 4and Corollary 1 above the desired inequality. The case when 1≤ q≤−p may be dealtwith similarly.(b). On combining the results of Theorem 2 and Theorem 3 we obtain the desiredinequality.(c). On combining the results of Corollary 1 and Corollary 2 we obtain the desiredinequality.

REMARK 2. Negative weighted power means of positive operators can be de-fined by a suitable limiting process as described in [2]. Since the details are no dif-ferent, they are hence not provided. All the inequalities established above hold forpositive operators.

References

[1] A LI c M., MOND B., PEcARI c J. AND VOLENEC V., Bounds for the differences of matrix means,SIAM J. Matrix Anal. Appl.18 (1) (1997), 119–123.

[2] BHAGWAT K.V. AND SUBRAMANIAN R., Inequalities between means of positive operators, Proc.Camb. Phil. Soc., (1978), 393–401.

[3] BHATIA R.,Matrix analysis, Springer-Verlag, 1997.

[4] NUSSBAUM R.D. AND COHEN J.E.,The arithmetic geometric mean and its generalizations for non-commuting linear operators, Ann. Sci. Norm. Sup. Sci. IV15 (2) (1989), 239–308.

[5] TRAPP G.E.,Hermition semidefinite matrix means and related inequalities, Linear and MultilinearAlg. 16 (1984), 113–123.

AMS Subject Classification: 47A63, 47A64

VASUDEVA H.L., Department of Mathematics, Panjab UniversityChandigarh-160014, INDIA.SINGH Mandeep, Department of Mathematics, S.L.I.E.T.Longowal-148106, INDIA.e-mail: [email protected]



N. Ujevic

AN APPLICATION OF THE MONTGOMERY IDENTITY TO

QUADRATURE RULES

Abstract. We use the Montgomery identity to obtain an optimal quadrature rule. It turns outthat this rule is the well-known compound trapezoidal rule.

1. Introduction

The Montgomery identity is recently considered by many authors ([1]-[5]). This iden-tity has the form

f (x) =1

b−a

Z b

af (t)dt +

Z b

aK(x,t) f ′(t)dt,

where

K(t) =

t −a, t ∈ [a,x]t −b, t ∈ (x,b]

.

From this identity we can obtain the well known integral Ostrowski inequality

∣∣∣∣ f (x)−1

b−a

Z b

af (t)dt

∣∣∣∣≤[

14

+(x− a+b

2 )2

(b−a)2

]∥∥ f ′∥∥

∞ (b−a).

This inequality is considered in really many recently published papers. It gives an errorbound for a simple quadrature rule.

In this paper we use the Montgomery identity and derive an optimal quadraturerule with respect to a given way of estimation of its error. Itturns out that this optimalquadrature rule is in fact the well-known compound trapezoid rule. In such a waywe proved that the compound trapezoidal rule is the best possible in a given class ofquadrature rules if we a priory give the way of estimation of remainder terms for theserules. In this sense the results of this paper can be considered as a generalization ofresults obtained in [6].

2. Quadrature rules

First we consider a problem in the interval[0,1]. Let 0 = x0 < x1 < · · · < xn = 1 bea partition of[0,1] and

(1)n

∑i=0

wi = 1, wi ∈ R.

137

138 N. Ujevic

If xi ∈ [0,1] then, using the Montgomery identity, we have

(2) f (xi) =Z 1

0f (t)dt+

Z 1

0K(xi ,t) f ′(t)dt,

whereK0(t) = t −1, t ∈ [0,1] ,

Ki(t) =

t, t ∈ [0,xi ]

t −1, t ∈ (xi ,1],

for i = 1,2, ...,n−1,xi ∈ [0,1] and

Kn(t) = t, t ∈ [0,1] .

From (2) we getn

∑i=0

wi f (xi) =

Z 1

0f (t)dt +

Z 1

0K(t) f ′(t)dt,

where

(3) K(t) =n

∑i=0

wiK(xi ,t).

The functionK(t) can be written in the equivalent form

K(t) =

t −w0, t ∈ [0,x1]

t −1∑j=0

wj , t ∈ (x1,x2]

t −2∑j=0

wj , t ∈ (x2,x3]

· · ·

t −n−1∑j=0

wj , t ∈ (xn−1,1]

.

We have ∣∣∣∣∣

Z 1

0f (t)dt−

n

∑i=0

wi f (xi)

∣∣∣∣∣≤∥∥ f ′∥∥

∞

Z 1

0|K(t)|dt.

We now consider the minimizing problem

(4)Z 1

0|K(t)|dt → min.

If we introduce the notation

σi =i

∑j=0

wj

Montgomery identity 139

then the problem (4) can be written in the form

(5)Z x1

0|t −σ0|dt+ · · ·+

Z 1

xn−1

|t −σn−1|dt → min.

We required thatσi ∈ [xi ,xi+1], i = 0,1, ...,n−1. Then we haveZ xi+1

xi

|t −σi|dt =

Z σi

xi

(σi − t)dt+Z xi+1

σi

(t −σi)dt(6)

=12

[(σi −xi)

2 +(xi+1−σi)2].

Using the relation (6) we can write the problem (5) in the form

F(w0, ...,wn−1) =12

n−1

∑i=0

[(σi −xi)

2 +(xi+1−σi)2]→ min.

We now solve this problem. For that purpose we calculate

(7)∂F∂wj

=n−1

∑i=0

[(σi −xi)

∂σi

∂wj− (xi+1−σi)

∂σi

∂wj

],

for j = 0,1,2, ...,n−1. Since

∂σi

∂wj=

1, j ≤ i0, j > i

from (7) we get∂F∂w0

=n−1

∑i=0

[2σi − (xi +xi+1)]

∂F∂w1

=n−1

∑i=1

[2σi − (xi +xi+1)]

· · ·

∂F∂wn−1

= 2σn−1− (xn−1+xn).

If we wish to find the minimum we have to require that

∂F∂wj

= 0, j = 0,1,2, ...,n−1.

We also introduce the notations

p j =n−1

∑i= j

(xi +xi+1),

q j =n−1

∑i= j

σi ,

140 N. Ujevic

for j = 0,1,2, ...,n−1. The we can write the above system in the form

2q0− p0 = 0

2q1− p1 = 0

· · ·

2qn−1− pn−1 = 0.

Since

p j − p j+1 = x j +x j+1,

q j −q j+1 = σ j ,

we have

2σ0− (x0+x1) = 0

2σ1− (x1+x2) = 0

· · ·

2σn−1− (xn−1+xn) = 0.

It is not difficult to solve the above system. We have

σ0 = w0 =x0 +x1

2,

σ1 = w0 +w1 =x1 +x2

2such that

w1 =x2−x0

2.

Generally, we have

wj =x j+1−x j−1

2, j = 2,3, ..,n−1,

and

wn =1−xn−1

2,

since (1) holds.

In fact, we get the next formula

Z 1

0f (t)dt =

n

∑i=0

wi f (xi)−Z 1

0K(t) f ′(t)dt,

whereK(t) is given by (3) andwi are given above. Hence, we got the following result.

THEOREM 1. Let f ∈C1(0,1). Then

(8)Z 1

0f (t)dt =

h0 f (0)+hn−1 f (1)

2+

n−1

∑i=1

hi−1+hi

2f (xi)+R( f ),


where

R( f ) = −

Z 1

0K(t) f ′(t)dt,

K(t) is given by (3), hi = xi+1−xi, i = 0,1,2, ...,n−1 and

(9) |R( f )| ≤‖ f ′‖∞

4

n−1

∑i=0

h2i .

Proof. The relation (8) is proved above. We have

Z 1

0|K(t)|dt =

n−1

∑i=0

[Z σi

xi

(σi − t)dt+Z xi+1

σi

(t −σi)dt

]

=12

n−1

∑i=0

[(σi −xi)

2 +(xi+1−σi)2]

=n−1

∑i=0

(xi+1−xi)2

4,

sinceσi =

xi +xi+1

2, i = 0,1,2, ...,n−1.

Hence, the relation (9) holds, too.

COROLLARY 1. Let f ∈C1(0,1). Then

(10)Z 1

0f (t)dt =

f (0)+ f (1)

2h+h

n−1

∑i=1

f (xi)+R1( f ),

where

R1( f ) = −

Z 1

0K(t) f ′(t)dt,

K(t) is given by (3), h= xi+1−xi = 1/n, i = 0,1,2, ...,n−1 and

|R1( f )| ≤h2

4

∥∥ f ′∥∥

∞ =‖ f ′‖∞

4n.

REMARK 1. Note that the quadrature rule given in Corollary 1 is the well-known compound trapezoidal quadrature rule. From the aboveconsiderations we canalso conclude that this rule is optimal in the sense that it has a minimal error boundwhen we estimate this error in the described way.

In fact, in both above cases we can obtain a better estimationof the error.

THEOREM2. Let the assumptions of Theorem 1 and Corollary 1 hold. Ifγ,Γ ∈R are numbers such thatγ ≤ f ′(t) ≤ Γ, t ∈ [0,1] then

(11) |R( f )| ≤Γ− γ

8

n−1

∑i=0

h2i ,

142 N. Ujevic

(12) |R1( f )| ≤h2

8(Γ− γ)

Proof. We have

Z 1

0K(t)dt =

Z x1

0(t −σ0)dt+ · · ·+

Z 1

xn−1

(t −σn−1)dt

=12

n−1

∑i=0

[(σi −xi+1)

2− (xi −σi)2]

= 0,

sinceσi =

xi +xi+1

2, i = 0,1,2, ...,n−1.

Thus,Z 1

0K(t)( f ′(t)−

Γ+ γ2

)dt =

Z 1

0K(t) f ′(t)

andZ 1

0

∣∣∣∣K(t)( f ′(t)−Γ+ γ

2)

∣∣∣∣≤Γ− γ

2

Z 1

0|K(t)|dt,

since ∥∥∥∥ f ′−Γ+ γ

2

∥∥∥∥∞≤

Γ− γ2

.

Now it is not difficult to see that (11) and (12) hold.

If we write the above rule in the interval[a,b] then we get the next result.

THEOREM 3. Let f ∈C1(a,b). Then

Z b

af (t)dt =

h0 f (a)+hn−1 f (b)

2+

n−1

∑i=1

hi−1+hi

2f (xi)+R( f ),

where hi = xi+1−xi, i = 0,1,2, ...,n−1 and

|R( f )| ≤Γ− γ

8

n−1

∑i=0

h2i ,

for γ,Γ ∈ R such thatγ ≤ f ′(t) ≤ Γ, t ∈ [a,b].

References

[1] A GLI C ALJINOVI C A., PECARIC J. AND VUKELI C A., The extension of Montgomery identity viaFink identity with applications, J. Inequal. Appl.1 (2005), 67–80.

[2] A GLI C ALJINOVI C A. AND PECARIC J., Discrete weighted Montgomery identity and discrete Os-trowski type inequalities, Comput. Math. Appl.48 (2004), 731–745.

[3] A NASTASSIOU G.,Univariate Ostrowski Inequalities, Revisited, Monatshefte fur Mathematik135(3)(2002), 175–189.


[4] CERONE P.AND DRAGOMIR S. S.,On some inequalities arising from Montgomery’s identity (Mont-gomery’s identity), J. Comput. Anal. Appl.5 (4) (2003), 341–367.

[5] PACHPATTE B., On Chebysev-Gruss type inequalities via Pecaric’s extension of the Montgomery iden-tity, J. Inequal. Pure Appl. Math.7 (1) (2006), 1–4.

[6] UJEVIC N., An optimal quadrature formula of closed type, Yokohama Math. J.50 (2003), 59–70.

AMS Subject Classification: 41A55.

Nenad UJEVIC, Department of Mathematics, University of Split, Teslina12/III, 21000 Split, CROATIAe-mail: [email protected]



E. Ballico∗

RANK 2 ARITHMETICALLY COHEN-MACAULAY VECTOR

BUNDLES ON CERTAIN RULED SURFACES

Abstract. Here we study rank 2 arithmetically Cohen-Macaulay vector bundles on a a ruledsurface over a smooth genusq curve, essentially proving their non-existence ifq≥ 2 and theruled surface is rather balanced.

1. Introduction

Let X be an integraln-dimensional projective variety,n ≥ 2, defined over an alge-braically closed field. Letη+ denote the ample cone of Pic(X) andη− its opposite. Letη0 (resp.η0) denote the set of all line bundles onX algebraically equivalent toOX (resp.numerically trivial). Setη := η+ ∪η−, γ := η∪η0 andγ := η∪ η0. Let E be a vectorbundle onX. We will say thatE is ACM or arithmetically Cohen-Macaulay(resp. saythatE is WACM or weakly arithmetically Cohen-Macaulay, resp. SACM orstronglyarithmetically Cohen-Macaulay) if H i(X,E⊗L) = 0 for all 1≤ i ≤ n−1 and allL ∈ γ(resp. L ∈ η, resp. L ∈ γ). Let C be a smooth and connected projective curve. Setq := pa(C). For any rank 2 vector bundleF onC sets(F) = deg(F)−2·deg(L), whereL is a maximal degree rank 1 subsheaf ofF . HenceF is stable (resp. semistable, resp.properly semistable) if and only ifs(F) > 0 (resp.s(F) ≥ 0, resp.s(F) = 0). A theo-rem of C. Segre and M. Nagata says thats(F) ≤ q. If s(F) ≥ 0, then sete(F) := s(F).If s(F) < 0, then sete(F) := 0.

THEOREM 1. Let C be a smooth curve of genus q≥ 2 and G a rank2 vectorbundle on C such that2q−3≥ max0,−s(G)+3e(G). Set X:= P(G). If q≥ 2, thenthere is no rank2 WACM vector bundle on X.

Of course, we will also check the rank 1 case (see Proposition1). As obviousfrom that proof and the proof of Theorem 1 with no restrictionon G there are verystrong numerical restrictions for the WACM and ACM line bundles and rank 2 vectorbundles on the ruled surfaceX. We stress the existence of rank 2 ACM vector bundleson X whenq = 1 andG = O ⊕2

C ([1]) and of rank one ACM line bundles whenq = 0,i.e. for Hirzebruch surfaces ([2]). For largee there are more (but always finitely many)isomorphism classes of line bundles onFe ([2]).

∗The author was partially supported by MIUR and GNSAGA of INdAM (Italy).

145

146 E. Ballico

2. The proof and related results

Notice that on any scroll over a smooth curve numerical equivalence and algebraicequivalence are the same. Henceη0 = η0 and γ = γ on any scroll over any smoothcurve.

REMARK 1. LetC be a smooth curve of genusq andF a rankr vector bundle onC. If h0(C,F ⊗L) = 0 for all L ∈ Pic0(X), then deg(F) < (r −1)(q−1) ([4], Corollaryat p. 252). Thus Riemann-Roch and Serre duality give that ifh1(C,F ⊗L) = 0 for allL ∈ Pic0(C), then deg(F) > (r +1)(q−1).

REMARK 2. Fix t ∈ Z. Fix a rank 2 vector bundleF on C. Setd := deg(F)ands := s(F). Let L be a maximal degree rank one subsheaf ofF . F/L is locally free,deg(L) = (d−2s)/2 and deg(F/L) = (d +2s)/2. Hences≡ d (mod 2). s(F ⊗R) =s(F) for all R∈ Pic(C). h0(C,F ⊗M) = 0 for all M ∈ Pict(C) if and only deg(L)+ t ≤−1, i.e. if and only if(d− 2s)/2+ t ≤ −1. Notice thats(F∗) = s(F). Hence Serreduality shows thath1(C,F ⊗M) = 0 for everyM ∈ Pict(C) if and only deg(F/L)+ t ≥2q−1, i.e. if and only if(d+2s)/2+ t ≥ 2q−1.

NOTATION 1. Fix a smooth and connected curveC with genusq and the ruledsurfaceX = P(G), whereG is a rank 2 vector bundle onC. LetG1 be a rank 1 subsheafof G. SinceG1 has maximal degree,G2 := G/G1 is a line bundle. Setai := deg(Gi).Hence deg(G) = a1 + a2 ands(G) = a2 − a1. SinceP(G) ∼= P(G⊗R) for any R∈Pic(C), we will always normalizeG so thatG2 ∼= OC. Hencea2 = 0, deg(G) = a1

ands(G) = −a1. Recall thate(G) := 0 if a1 ≥ 0 ande(G) := −a1 if a1 < 0. Noticethar 0≤ e(G) ≤ q for anyX (Remark 2). Letπ : X → C denote the ruling andOπ(1)the tautologicalπ-ample line bundle onX. Pic(X) ∼= ZOπ(1)⊕π∗(Pic(C)). For everyintegert and everyM ∈ Pic(C) setOπ(t) := O ⊗t

π andOX(t,M) := Oπ(t)⊗π∗(M).

REMARK 3. TakeC,G,X,a1,e(G) as in Notation 1. FixD ∈ Pic(C). NoticethatX ∼= P(G⊗D). Applying [3], Theorem III.1.1, to the vector bundleG⊗D we getthatOX(1,D) is ample if and only if deg(D) ≥ 1+e(G).

First Claim: For every integerx > 0 Sx(G)⊗D is an ample vector bundle ifdeg(D) ≥ 1+xe(G).

Proof of the First Claim: The vector bundleSx(G) has rankx+ 1 and ithas an increasing filtrationFi0≤i≤x such thatF0 = 0, Fx+1 = Sx(G), eachFi/Fi−1,1≤ i ≤ x+1, is a line bundle of degree≥ 0 (casee(G) = 0) or degree≥−xe(G) (casee(G) > 0), and deg(F1) = xa1. Just use that an extension of ample line bundles is ampleand that a line bundle onC is ample if it has positive degree.

Second Claim:Fix an integerx≥ 1 and assume deg(D) ≥ 1+ xe(G). ThenR := OX(x,D) is ample.

Proof of the Second Claim:By Nakai criterion ([3], Theorem I.5.1) it is suf-ficient to prove thatR2 > 0 and thatOX(T) ·R > 0 for every integral curveT ⊂ X.R2 = 2x·deg(D)+x2a1 > 0. Take an integral curveT ⊂ X and setOX(y,M) := OX(T).Notice thaty≥ 0 and thaty = 0 if and only if T is a fiber ofπ. OX(T) ·R= xya1 +x ·

vector bundles on ruled surfaces 147

deg(M)+y·deg(D). If y = 0, thenOX(T) ·R= x > 0. From now on we assumey > 0.First assumea1 ≥ 0. Hencee(G) = 0 andOX(T) ·R≥ xya1+x·deg(M)+y> x(ya1+deg(M)). Hence it is sufficient to prove that deg(M) ≥ −ya1. Assume deg(M) ≤−ya1−1. To get a contradiction it is sufficient to show thath0(X,OX(y,M)) = 0. Sincey> 0,h0(X,OX(y,M)) = h0(C,Sy(G)⊗M). The vector bundleSy(G) has ranky+1 andit has an increasing filtrationFi0≤i≤x such thatF0 = 0, Fy+1 = Sy(G), eachFi/Fi−1,1 ≤ i ≤ y+ 1, is a line bundle of degree(y+ 1− i)a1. Henceh0(X,OX(y,M)) = 0.Now assumea1 < 0. Hencee(G) = −a1 andOX(T) ·R≥ y+ x ·deg(M). Hence it issufficient to observe that the same filtration ofSy(G) used in the previous case givesh0(C,Sy(G)⊗M) = 0 if deg(M) < 0.

REMARK 4. TakeC,G,X,a1,e(G) as in Notation 1. LetF be a rank 2 vectorbundle onC.

(a) SetE := π∗(F). We want to check thatE is not WACM if 3e(G)≤ 2q−3.Assume thatE is WACM. h1(X,E(1,D)) = h1(C,G⊗F ⊗D). If h1(C,G⊗F ⊗D) =0, thenh1(C,G2 ⊗ F ⊗D) = 0. Recall thatG2 ∼= OC and thatOX(1,D) is ample ifdeg(D)≥ 1+e(G). VaryingD∈Pic1+e(D)(C) and applying Remark 1 we get deg(F)≥

3q−3−e(G). SetJ := OX(2,M) with M ∈ Pic1+2e(G)(C). J is ample. Serre dualitygivesh1(X,E ⊗ J∗) = h1(X,E∗(0,F∗ ⊗ωC ⊗ det(G)⊗M ⊗A∗)) = h1(C,F∗ ⊗ωC ⊗det(G)⊗M) = h0(C,F ⊗M∗). Remark 1 shows that ifh0(C,F ⊗M∗) = 0 for all M,then deg(F) ≤ q−2+1+2e(G).

(b) SetE := π∗(F)(−1,OC). h1(X,E⊗L) = 0 for all L ∈ η0. Here we checkthatE is not WACM if 3e(G)≤ 2q−2. Assume thatE is WACM. h1(X,E(1,D)) = 0 ifand only ifh1(X,F ⊗D) = 0. Hence we geth1(X,F ⊗D) = 0 for all D∈ Pic1+e(G)(C).Remark 1 gives deg(F)+ 2+ 2e(G) > 3(q−1), i.e. deg(F) ≥ 3q−2−2e(G). Serreduality shows thath1(X,E(−1,M)) = 0 if and only if h1(X,π∗(F∗)(0,M∗⊗ωC)) = 0,i.e. if and onlyh1(C,F∗⊗M∗⊗ωC) = 0, i.e. if and only ifh0(C,F ⊗M) = 0. VaryingM in Pic−1−e(G)(C) we get deg(F) ≤ q−1+e(G).

(c) SetE := π∗(F)(−2,OC). Serre duality and part (a) shows thatE is notWACM if 3e(G) ≤ 2q−3.

PROPOSITION1. Take C,G,X,a1,e(G) as in Notation 1. If q≥ 2 and2q−3≥max0,a1+3e(G), then there is no WACM line bundle on X.

Proof. Fix anyR := OX(x,A) ∈ Pic(X) and assume thatR is WACM.

(a) Here we assumex ≥ −1. Take anyL := OX(1,D) such that deg(D) =1+ e(G). L is ample (Remark 3). Sincex+ 1 ≥ 0, h1(X,R⊗ L) = 0 if and only ifh1(C,Sx+1(G)⊗A⊗D)= 0. SinceOC = G2 is a quotient ofG, OC is a quotient ofSt(G)for anyt > 0. Hence ift > 0,M ∈Pic(C) andh1(C,St(G)⊗M) = 0, thenh1(C,M) = 0.VaryingD in Pic1+e(G)(C) we see that ifR is WACM, then deg(A)+1+e(G)≥ 2q−1,i.e. deg(A) ≥ 2q−2−e(G).

(b) Here we assumex > 0. SetL := OX(x,D) with deg(D) ≫ 0. SinceL isample andh1(X,R⊗L∗) = h1(C,A⊗D∗) > 0 if deg(D) ≫ 0, R is not WACM.

(c) Here we assumex = 0. TakeL := OX(2,D) with deg(D) = 2 ·e(G)+ 1.

148 E. Ballico

HenceL is ample (Remark 4). Serre duality givesh1(X,R⊗L∗) = h1(X,OX(0,ωC⊗

det(G)⊗D⊗A∗)) = h1(C,ωC ⊗det(G)⊗D⊗⊗A∗). Varying D in Pic1+2e(G)(C) wesee thatR is not WACM if det(G)+1+2·e(G)−deg(A)≤ 0, i.e. if deg(A)≥ a1+1+2e(G). If 2q−2−e(G)≥ a1 +1+2e(G), then part (a) shows thatR is not WACM.

(d) Here we assumex = −1. TakeL := OX(1,D) with deg(D) = 1+ e(G).L is ample. h1(X,R⊗ L) = h0(C,A⊗D). Hence varyingD in Pic1+e(G)(C) we seethat if R is WACM, then deg(A) + 1+ e(G) ≥ 2q− 1, i.e. deg(A) ≥ 2q− 2− e(G).Serre duality givesh1(X,R⊗L∗) = h1(X,OX(0,D⊗A∗⊗ωC⊗det(G))). Hence IfRis WACM, then 1+ e(G)− deg(A)+ 2q− 2+ a1 ≥ 2q− 1, i.e. deg(A) ≤ e(G)+ a1.Thus if R is WACM, then 2q−2−e(G)≤ deg(A) ≤ e(G)+ a1. First assumea1 ≤ 0.Hencee(G) = −a1. Sinceq≥ 2, we get a contradiction. Now assumea1 > 0. Hencee(G) = 0. In this case the contradiction comes from the assumption 2q−1≥ a1.

(e) Here we assumex ≤ −2. Serre duality shows thatR is not WACM un-der the same assumptions we used in the casex ≥ 0. Notice that ifx < −2, then noassumption at all is needed.

Proof of Theorem 1. Let E be a rank 2 WACM vector bundle onX.Since Pic(X) ∼= ZOπ(1)⊕π∗(Pic(C)), there are an integerx andA∈ Pic(C) such thatdet(E) ∼= OX(x,A). By [1], proof of Theorem 2, and [2], Theorem 1,−4≤ x≤ 0 andthere are an integerz∈ −2,−1,0, N ∈ Pic(C), and an exact sequence

(1) 0→ OX(z,N) → E → OX(x−z,A⊗N∗) → 0

Moreover,x≤ 2z.

(a) Here we assumex = 2z. A base-change theorem ([5], p. 11) says thatF := π∗(E(−z,OC)) is a rank 2 vector bundle onC and that the natural mapπ∗(F) →E(−z,OC) is an isomorphism. Apply Proposition 1. Hence from now on in the proofwe will assumex < 2zand in particularz∈ −1,0.

(b) Here we assumez= −1. Hencex∈ −4,3. Fix anyD ∈ Pic1+e(G)(G)and setL := OX(1,D). L is ample (Remark 3). Sincex− z+ 1 < 0, h0(X,OX(x− z+1,A⊗N∗⊗D))= 0. SinceE is WACM, the exact sequence (1) givesh1(X,OX(−1,N)⊗L) = 0. Sincex−z−1< 0,h0(X,OX(x−z−1,A⊗N∗⊗D∗)) = 0. SinceE is WACM,we geth1(X,OX(−1,N)⊗L∗) = 0. Part (d) of the proof of Proposition 1 gives a con-tradiction, becauseq≥ 2 and 2q−1≥ a1.

(c) Here we assumez= 0 andx≤−2. TakeL as in part (b). Sinceh0(X,OX(x−z+ 1,A⊗N∗ ⊗D)) = h0(X,OX(x− z−1,A⊗N∗ ⊗D∗)) = 0, we conclude as in part(b).

(d) Here we consider the case(z,x) = (0,−1), i.e. the unique remainingcase. Fix anyD ∈ Pic1+e(G)(G) and setL := OX(1,D). L is ample (Remark 3). SetR := OX(−1,A⊗N∗). Sinceh2(X,OX(1,N⊗D)) = h2(X,OX(−1,N⊗D∗)) = 0 andEis WACM, the exact sequence (1) givesh1(X,R⊗L) = h1(X,R⊗L∗) = 0. Part (d) ofthe proof of Proposition 1 gives a contradiction, becauseq≥ 2 and 2q−1≥ a1.

vector bundles on ruled surfaces 149

References

[1] E. Ballico, ACM vector bundles on products of smooth curves, preprint.

[2] E. Ballico, ACM vector bundles on scrolls, preprint.

[3] R. Hartshorne, Ample subvarieties of algebraic varieties, Lect. Notes in Math. 156, Springer, Berlin,1970.

[4] S. Mukai and F. Sakai, Maximal subbundles of vector bundles on a curve, Manuscripta Math. 52 (1985),no. 1-3, 251–256.

[5] C. Okonek, M. Schneider and H. Spindler, vector bundles on projective spaces, Birkhauser, Boston,1980.

AMS Subject Classification: 14J60; 14H60.

E. Ballico, Dept. of Mathematics, University of Trento, 38050 Povo (TN), ITALYe-mail: [email protected]



L. Selmani - N. Bensebaa∗

AN ELECTRO-VISCOELASTIC CONTACT PROBLEM WITH

ADHESION AND DAMAGE

Abstract. We consider a quasistatic frictionless contact problem foran electro-viscoelasticbody with damage. The contact is modelled with normal compliance. The adhesion ofthe contact surfaces is taken into account and modelled by a surface variable, the bondingfield. We derive variational formulation for the model whichis in the form of a systeminvolving the displacement field, the electric potential field, the damage field and the adhesionfield. We prove the existence of a unique weak solution to the problem. The proof is basedon arguments of time-dependent variational inequalities,parabolic inequalities, differentialequations and fixed point.

1. Introduction

The piezoelectric phenomenon represents the coupling between the mechanical andelectrical behavior of a class of materials, called piezoelectric materials. In simplesttems, when a piezoelectric material is squeezed, an electric charge collects on its sur-face, conversely, when a piezoelectric material is subjected to a voltage drop, it me-chanically deforms. Many crystalline materials exhibit piezoelectric behavior. A fewmaterials exhibit the phenomenon strongly enough to be usedin applications that takeadvantage of their properties. These include quartz, Rochelle salt, lead titanate zir-conate ceramics, barium titanate and polyvinylidene flouride (a polymer film). Piezo-electric materials are used extensively as switches and actually in many engineer-ing systems in radioelectronics, electroacoustics and measuring equipment. However,there are very few mathematical results concerning contactproblems involving piezo-electric materials and therefore there is a need to extend the results on models forcontact with deformable bodies which include coupling between mechanical and elec-trical properties. General models for elastic materials with piezoelectric effects can befound in [11,12,13,21,22] and more recently in[1,20]. The adhesive contact betweendeformable bodies, when a glue is added to prevent relative motion of the surfaces, hasalso received recently increased attention in the mathematical literature. Analysis ofmodels for adhesive contact can be found in[3,4,6,7,15,16,17] and recently in themonographs[18,19] . The novelty in all these papers is the introduction of a surfaceinternal variable, the bonding field, denoted in this paper by α, which describes thepointwise fractional density of adhesion of active bonds onthe contact surface, and issometimes referred to as the intensity of adhesion. Following [6,7], the bonding fieldsatisfies the restriction 0≤ α ≤ 1, whenα = 1 at a point of the contact surface, the ad-hesion is complete and all the bonds are active, whenα = 0 all the bonds are inactive,severed, and there is no adhesion, when 0< α < 1 the adhesion is partial and only afractionα of the bonds is active.

The importance of this paper is to make the coupling of an electro-viscoelastic

151

152 L. Selmani - N. Bensebaa

problem with damage and a frictionless contact problem withadhesion. We study aquasistatic problem of frictionless adhesive contact. We model the material behaviorwith an electro-viscoelastic constitutive law with damageand the contact with normalcompliance with adhesion. We derive a variational formulation and prove the existenceand uniqueness of the weak solution.

The paper is structured as follows. In section 2 we present notation and somepreliminaries. The model is described in section 3 where thevariational formulation isgiven. In section 4, we present our main result stated in Theorem 2 and its proof whichis based on arguments of time-dependent variational inequalities, parabolic inequali-ties, differential equations and fixed point.

2. Notation and preliminaries

In this short section, we present the notation we shall use and some preliminary mate-rial. For more details, we refer the reader to[2,5,14] . We denote bySd the space ofsecond order symmetric tensors onRd (d = 2,3), while ”.” and | . | represent the innerproduct and the Euclidean norm onSd andRd, respectively. We recall that the innerproducts and the corresponding norms onRd andSd are given by

u . v = uivi , | v | = (v,v)12 ∀u,v ∈R

d,

σ . τ = σi j τi j , | τ |= (τ,τ)12 ∀σ,τ ∈Sd,

respectively. Here and below, the indicesi and j run from 1 tod, the summationconvention over repeated indices is used and the index that follows a comma indicatesa partial derivative with respect to the corresponding component of the independentvariable.

Let Ω ⊂ Rd be a bounded domain with a regular boundaryΓ and letν denotethe unit outer normal onΓ. We shall use the notation

H = L2(Ω)d =

u = (ui) / ui ∈ L2(Ω)

,

H1(Ω)d =

u = (ui) / ui ∈ H1(Ω)

,

H =

σ = (σi j ) / σi j = σ ji ∈ L2(Ω)

,

H1 =

σ ∈ H / Div σ ∈ H

,

we consider thatε : H1(Ω)d →H andDiv :H1 →H are the deformation and divergenceoperators, respectively, defined by

ε(u) = (εi j (u)), εi j (u) =12(ui, j +u j ,i), Divσ = (σi j , j).

The spacesH, H1(Ω)d, H andH1 are real Hilbert spaces endowed with the canonicalinner products given by

(u,v)H =

Z

Ωu . v dx ∀u,v ∈ H,

Contact problem with adhesion and damage 153

(u,v)H1(Ω)d =

Z

Ωu . v dx+

Z

Ω∇u . ∇v dx ∀u,v ∈ H1(Ω)d,

where∇v = (vi, j) ∀v ∈ H1(Ω)d,

(σ,τ)H =Z

Ωσ . τ dx ∀σ,τ ∈ H ,

(σ,τ)H1= (σ,τ)H +(Div σ,Div τ)H ∀σ,τ ∈ H1.

The associated norms on the spacesH, H1(Ω)d, H andH1 are denoted by| . |H ,

| . |H1(Ω)d , | . |H and| . |H1respectively. LetHΓ = H

12 (Γ)d and letγ : H1(Ω)d → HΓ be

the trace map. For every elementv ∈ H1(Ω)d, we also use the notationv to denote thetraceγv of v onΓ and we denote byvν andvτ the normal and the tangential componentsof v on the boundaryΓ given by

(1) vν = v . ν, vτ = v−vνν.

We note thatvν is a scalar, whereasvτ is a tangent vector toΓ. In particular, in whatfollows, uν anduτ will represent the normal and tangential displacement. Similarly,for a regular (sayC1) tensor fieldσ : Ω → Sd we define its normal and tangentialcomponents by

(2) σν = (σν) . ν, στ = σν−σνν,

and we recall that the following Green’s formula holds :

(σ,ε(v))H +(Div σ,v)H =

Z

Γσν . v da ∀v ∈ H1(Ω)d.

Finally, for any real Hilbert spaceX, we use the classical notation for the spacesLp(0,T;X) andWk,p(0,T;X), where 1≤ p≤ +∞ andk≥ 1. We denote byC(0,T;X)andC1(0,T;X) the space of continuous and continuously differentiable functions from[0,T] to X, respectively, with the norms

| f |C(0,T;X)= maxt∈[0,T]

| f(t) |X ,

| f |C1(0,T;X)= maxt∈[0,T]

| f(t) |X + maxt∈[0,T]

|.f (t) |X ,

respectively. Moreover, we use the dot above to indicate thederivative with respect tothe time variable and, for a real numberr, we user+ to represent its positive part, thatis r+ = max0, r. For the convenience of the reader, we recall the following versionof the classical theorem of Cauchy-Lipschitz (see, e.g.,[19,p.48]).

THEOREM 1. Assume that(X, | . |X) is a real Banach space and T> 0. LetF(t, .) : X →X be an operator defined a.e. on(0,T) satisfying the following conditions:


1 - There exists a constantLF > 0 such that

| F(t,x)−F(t,y) |X≤ LF | x−y |X ∀x,y∈ X, a.e.t ∈ (0,T) .

2 - There existsp≥ 1 such thatt 7−→ F(t,x) ∈ Lp(0,T;X) ∀x∈ X.

Then for anyx0 ∈ X, there exists a unique functionx∈W1, p(0,T;X) such that

.x (t) = F(t,x(t)) a.e. t ∈ (0,T) ,

x(0) = x0.

Theorem 1 will be used in section 4 to prove the unique solvability of the intermediateproblem involving the bonding field. Moreover, ifX1 andX2 are real Hilbert spaces thenX1×X2 denotes the product Hilbert space endowed with the canonical inner product(., .)X1×X2.

3. Mechanical and variational formulations

We describe the model for the process, we present its variational formulation. Thephysical setting is the following. An electro-viscoelastic body occupies a bounded do-mainΩ ⊂ R

d (d = 2,3) with outer Lipschitz surfaceΓ. The body undergoes the actionof body forces of densityf0 and volume electric charges of densityq0. It also undergoesthe mechanical and electric constraint on the boundary. We consider a partition ofΓinto three disjoint measurable partsΓ1, Γ2 andΓ3, on one hand, and in two measurablepartsΓa andΓb, on the other hand, such thatmeas(Γ1) > 0, meas(Γa) > 0 andΓ3

⊂ Γb. Let T > 0 and let[0,T] be the time interval of interest. The body is clamped onΓ1× (0,T), so the displacement field vanishes there. A surface tractions of densityf2

act onΓ2×(0,T) and a body force of densityf0 acts inΩ×(0,T) . We also assume thatthe electrical potential vanishes onΓa× (0,T) and a surface electric charge of densityq2 is prescribed onΓb × (0,T). The body is in adhesive contact with an obstacle, orfoundation, over the contact surfaceΓ3. We suppose that the body forces and tractionsvary slowly in time, and therefore, the accelerations in thesystem may be neglected.Neglecting the inertial terms in the equation of motion leads to a quasistatic approachof the process. We denote byu the displacement field, byσ the stress tensor field andby ε(u) the linearized strain tensor. We use an electro-viscoelastic constitutive law withdamage given by

σ = A ε(.u)+G (ε(u),β)−E ∗E(ϕ),

D = E ε(u)+BE(ϕ),

whereA is a given nonlinear function,E(ϕ) = −∇ϕ is the electric field,E = (ei jk)represents the third order piezoelectric tensor,E ∗ is its transpose andB denotes theelectric permittivity tensor.G represents the elasticity operator whereβ is an internalvariable describing the damage of the material caused by elastic deformations. Thedifferential inclusion used for the evolution of the damagefield is

.

β −kβ + ∂ϕK(β) ∋ S(ε(u),β),


whereK denotes the set of admissible damage functions defined by

K = ξ ∈ H1(Ω) / 0≤ ξ ≤ 1 a.e. in Ω,

k is a positive coefficient,∂ϕK denotes the subdifferential of the indicator functionϕK

andS is a given constitutive function which describes the sources of the damage in thesystem. Whenβ = 1 the material is undamaged, whenβ = 0 the material is completelydamaged, and for 0< β < 1 there is partial damage. General models of mechanicaldamage, which were derived from thermodynamical considerations and the principle ofvirtual work, can be found in[8] and[9] and references therein. The models describe theevolution of the material damage which results from the excess tension or compressionin the body as a result of applied forces and tractions. Mathematical analysis of one-dimensional damage models can be found in[10] .

To simplify the notation, we do not indicate explicitly the dependence of variousfunctions on the variablesx ∈ Ω∪Γ andt ∈ [0,T] . Then, the classical formulation ofthe mechanical problem of electro-viscoelastic material,frictionless, adhesive contactmay be stated as follows.

Problem P. Find a displacement fieldu : Ω× [0,T] → Rd, an electric potentialfield ϕ : Ω× [0,T] → R, a damage fieldβ : Ω× [0,T] → R and a bonding fieldα: Γ3× [0,T] → R such that

(3) σ = A ε(.u)+G (ε(u),β)+E ∗∇ϕ in Ω× (0,T) ,

(4) D = E ε(u)−B∇ϕ in Ω× (0,T) ,

(5).

β −kβ + ∂ϕK(β) ∋ S(ε(u),β) in Ω× (0,T) ,

(6) Div σ+ f0 = 0 in Ω× (0,T) ,

(7) div D = q0 in Ω× (0,T) ,

(8) u = 0 on Γ1× (0,T) ,

(9) σν = f2 on Γ2× (0,T) ,

(10) −σν = pν(uν)− γνα2Rν(uν) on Γ3× (0,T) ,

(11) −στ = pτ(α)Rτ(uτ) on Γ3× (0,T) ,

(12).α= −

(α(γν(Rν(uν))

2 + γτ | R τ(uτ) |2 )− εa

)+

on Γ3× (0,T) ,


(13)∂β∂ν

= 0 on Γ× (0,T) ,

(14) ϕ = 0 on Γa× (0,T) ,

(15) D . ν = q2 on Γb× (0,T) ,

(16) u(0) = u0,β(0) = β0 in Ω,

(17) α(0) = α0 on Γ3.

First, 3 and 4 represent the electro-viscoelastic constitutive law with damage,the evolution of the damage field is governed by the inclusionof parabolic type givenby the relation 5, whereS is the mechanical source of the damage growth, assumed tobe rather general function of the strains and damage itself,∂ϕK is the subdifferentialof the indicator function of the admissible damage functions setK. Equations 6 and 7represent the equilibrium equations for the stress and electric-displacement fields while8 and 9 are the displacement and traction boundary condition, respectively. Condition10 represents the normal compliance condition with adhesion whereγν is a given ad-hesion coefficient andpν is a given positive function which will be described below.In this condition the interpenetrability between the body and the foundation is allowed,that isuν can be positive onΓ3. The contribution of the adhesive to the normal tractionis represented by the termγνα2Rν(uν), the adhesive traction is tensile and is propor-tional, with proportionality coefficientγν, to the square of the intensity of adhesion andto the normal displacement, but as long as it does not exceed the bond lengthL. Themaximal tensile traction isγνL. Rν is the truncation operator defined by

Rν(s) =

L if s< −L,−s if −L ≤ s≤ 0,0 if s> 0.

HereL > 0 is the characteristic length of the bond, beyond which it does notoffer any additional traction. The introduction of the operatorRν, together with the op-eratorRτ defined below, is motivated by mathematical arguments but itis not restrictivefrom the physical point of view, since no restriction on the size of the parameterL ismade in what follows. Condition 11 represents the adhesive contact condition on the


tangential plane, in whichpτ is a given function andRτ is the truncation operator givenby

Rτ(v) =

v if | v | ≤ L,L v|v| if | v | > L.

This condition shows that the shear on the contact surface depends on the bond-ing field and on the tangential displacement, but as long as itdoes not exceed the bondlengthL. The frictional tangential traction is assumed to be much smaller than theadhesive one and, therefore, omitted.

Next, the equation 12 represents the ordinary differentialequation which de-scribes the evolution of the bonding field and it was already used in[3], see also[18,19]for more details. Here, besidesγν, two new adhesion coefficients are involved,γτ andεa. Notice that in this model once debonding occurs bonding cannot be reestablishedsince, as it follows from 12,

.α≤ 0. The relation 13 represents a homogeneous Neu-

mann boundary condition where∂β∂ν represents the normal derivative ofβ. 14 and 15

represent the electric boundary conditions. 16 representsthe initial displacement fieldand the initial damage field. Finally 17 represents the initial condition in whichα0 isthe given initial bonding field. To obtain the variational formulation of the problem3-17, we introduce for the bonding field the set

Z =

θ ∈C(0,T;L2(Γ3)) / 0≤ θ(t) ≤ 1 ∀t ∈ [0,T] , a.e. onΓ3

,

and for the displacement field we need the closed subspace ofH1(Ω)d defined by

V =

v ∈ H1(Ω)d / v = 0 onΓ1

.

Sincemeas(Γ1) > 0, Korn’s inequality holds and there exists a constantCk > 0, thatdepends only onΩ andΓ1, such that

| ε(v) |H ≥Ck | v |H1(Ω)d ∀v ∈V.

A proof of Korn’s inequality may be found in[14,p.79]. On the spaceV we considerthe inner product and the associated norm given by

(18) (u,v)V = (ε(u),ε(v))H , | v |V=| ε(v) |H ∀u,v ∈V.

It follows that | . |H1(Ω)d and | . |V are equivalent norms onV and therefore(V, | . |V) is a real Hilbert space. Moreover, by the Sobolev trace Theorem and 18, thereexists a constantC0 > 0, depending only onΩ, Γ1 andΓ3 such that

(19) | v |L2(Γ3)d≤C0 | v |V ∀v ∈V.

We also introduce the spaces

W =

φ ∈ H1(Ω) / φ = 0 onΓa

,


W =

D = (Di) / Di ∈ L2(Ω), div D ∈ L2(Ω)

,

wherediv D = (Di,i). The spacesW andW are real Hilbert spaces with the innerproducts given by

(ϕ,φ)W =Z

Ω∇ϕ . ∇φ dx,

(D,E)W =

Z

ΩD . E dx+

Z

Ωdiv D . div E dx.

The associated norms will be denoted by| . |W and| . |W , respectively. Moreover, whenD ∈W is a regular function, the following Green’s type formula holds:

(D,∇φ)H +(div D,φ)L2(Ω) =

Z

ΓD . ν φ da ∀φ ∈ H1(Ω).

Notice also that, sincemeas(Γa) > 0, the following Friedrichs-Poincare inequalityholds:

(20) | ∇φ |H≥CF | φ |H1(Ω) ∀φ ∈W,

whereCF > 0 is a constant which depends only onΩ and Γa. In the study of themechanical problem 3-17, we assume that the viscosity function A : Ω×Sd → Sd

satisfies

(21)

(a) There exists a constantLA > 0 Such that| A (x,ε1)−A (x,ε2) |≤ LA | ε1− ε2 | ∀ε1,ε2 ∈ Sd, a.e.x ∈ Ω.(b) There exists a constantmA > 0 Such that(A (x,ε1)−A (x,ε2)) . (ε1− ε2) ≥ mA | ε1− ε2 |

2 ∀ε1,ε2 ∈ Sd, a.e.x ∈ Ω.(c) The mappingx → A (x,ε) is Lebesgue measurable onΩ for anyε ∈ Sd.(d) The mappingx → A (x,0) belongs toH .

The elasticity OperatorG : Ω×Sd×R → Sd satisfies

(22)

(a) There exists a constantLG > 0 Such that| G (x,ε1,α1)−G (x,ε2,α2) |≤ LG (| ε1− ε2 | + | α1−α2 |)∀ε1,ε2 ∈ Sd, ∀α1,α2 ∈ R a.e.x ∈ Ω.(b) The mappingx → G (x,ε,α) is Lebesgue measurable onΩfor anyε ∈ Sd andα ∈ R.(c) The mappingx → G (x,0,0) belongs toH .

The damage source functionS : Ω×Sd×R → R satisfies(23)

(a) There exists a constantLS > 0 such that| S(x,ε1,α1)− S(x,ε2,α2) |≤ LS(| ε1− ε2 | + | α1−α2 |)∀ε1,ε2 ∈ Sd, ∀α1,α2 ∈ R a.e.x ∈ Ω.

(b) For anyε ∈ Sd andα ∈ R, x → S(x,ε,α) is Lebesgue measurable onΩ.(c)The mappingx → S(x,0,0) belongs toL2(Ω).


The electric permittivity operatorB = (bi j ): Ω×Rd → Rd satisfies

(24)

(a) B(x,E) = (bi j (x)E j) ∀E = (Ei) ∈ Rd, a.e.x ∈ Ω.(b) bi j = b ji , bi j ∈ L∞(Ω), 1≤ i, j ≤ d.(c) There exists a constantmB > 0 such thatBE.E ≥ mB | E |2 ∀E = (Ei) ∈ Rd, a.e. inΩ.

The piezoelectric operatorE : Ω×Sd → Rd satisfies(25)

(a) E (x,τ)=(ei j k (x)τ jk) ∀τ = (τi j ) ∈ Sd, a.e.x ∈ Ω.

(b) ei jk = eik j ∈ L∞(Ω), 1≤ i, j,k ≤ d.

The normal compliance functionpν : Γ3×R → R+ satisfies

(26)

(a) There exists a constantLν > 0 such that| pν(x, r1)− pν(x, r2) |≤ Lν | r1− r2 | ∀r1, r2 ∈ R, a.e.x ∈ Γ3.(b) The mappingx → pν(x, r) is measurable onΓ3, for anyr ∈ R.(c) pν(x, r) = 0 for all r ≤ 0, a.e.x ∈ Γ3.

The tangential contact functionpτ : Γ3×R → R+ satisfies

(27)

(a) There exists a constantLτ > 0 such that| pτ(x,d1)− pτ(x,d2) |≤ Lτ | d1−d2 | ∀d1,d2 ∈ R, a.e.x ∈ Γ3.(b) There existsMτ > 0 such that| pτ(x,d) |≤ Mτ ∀d ∈ R,a.e.x ∈ Γ3.(c) The mappingx → pτ(x,d) is measurable onΓ3, for anyd ∈ R.(d) The mappingx → pτ(x,0) ∈ L2(Γ3).

We also suppose that the body forces and surface tractions have the regularity

(28) f0 ∈C(0,T;H), f2 ∈C(0,T;L2(Γ2)d),

(29) q0 ∈C(0,T;L2(Ω)), q2 ∈C(0,T;L2(Γb)).

(30) q2(t) = 0 onΓ3 ∀t ∈ [0,T] .

Note that we need to impose assumption 30 for physical reasons, indeed the foundationis assumed to be insulator and therefore the electric charges (which are prescribed onΓb ⊃ Γ3) have to vanish on the potential contact surface. The adhesion coefficientssatisfy

(31) γν,γτ ∈ L∞(Γ3),εa ∈ L2(Γ3),γν,γτ,εa ≥ 0 a.e. onΓ3.

The initial displacement field satisfies

(32) u0 ∈V,


the initial bonding field satisfies

(33) α0 ∈ L2(Γ3), 0≤ α0 ≤ 1 a.e. onΓ3,

and the initial damage field satisfies

(34) β0 ∈ K.

We define the bilinear forma : H1(Ω)×H1(Ω) → R by

(35) a(ξ,ϕ) = kZ

Ω∇ξ . ∇ϕ dx.

Next, we denote byf : [0,T] →V the function defined by

(36) (f(t),v)V =Z

Ωf0(t) . v dx +

Z

Γ2

f2(t) . v da ∀v ∈V, t ∈ [0,T] ,

and we denote byq : [0,T] →W the function defined by

(37) (q(t),φ)W =

Z

Ωq0(t) . φ dx −

Z

Γb

q2(t) . φ da ∀φ ∈W, t ∈ [0,T] .

Next, we denote byj : L∞(Γ3)×V×V → R the adhesion functional defined by

(38) j(α,u,v) =

Z

Γ3

pν(uν)vν da+

Z

Γ3

(−γνα2Rν(uν)vν + pτ(α)Rτ(u τ) . vτ) da.

Keeping in mind 26 and 27, we observe that the integrals 38 arewell defined and wenote that conditions 28 and 29 imply

(39) f ∈C(0,T;V), q∈C(0,T;W).

Using standard arguments we obtain the variational formulation of the mechanicalproblem 3-17.

Problem PV. Find a displacement fieldu : [0,T]→V, an electric potential fieldϕ : [0,T] →W, a damage fieldβ : [0,T] → H1(Ω) and a bonding fieldα : [0,T] → L∞

(Γ3) such that

(A ε(.u (t)),ε(v))H +(G (ε(u(t)),β(t)),ε(v))H +(E ∗∇ϕ(t),ε(v))H

(40) + j(α(t),u(t),v) = (f(t),v)V ∀v ∈V,t ∈ (0,T) ,

β(t) ∈ K for all t ∈ [0,T] , (.

β (t),ξ−β(t))L2(Ω) +a(β(t),ξ−β(t))

(41) ≥ (S(ε(u(t)),β(t)),ξ−β(t))L2(Ω) ∀ξ ∈ K,


(42) (B∇ϕ(t),∇φ)H − (E ε(u(t)),∇φ)H = (q(t),φ)W ∀φ ∈W,t ∈ (0,T) ,

(43).α (t) = −

(α(t)(γν(Rν(uν(t)))

2 + γτ | R τ(uτ(t)) |2 )− εa

)+

a.e.t ∈ (0,T) ,

(44) u(0) = u0,β(0) = β0,α(0) = α0.

We notice that the variational problemPV is formulated in terms of displacement field,an electrical potential field, damage field and bonding field.The existence of the uniquesolution of problemPV is stated and proved in the next section. To this end, we con-sider the following remark which is used in different placesof the paper.

REMARK 1. We note that, in the problemP and in the problemPV we do notneed to impose explicitly the restriction 0≤ α ≤ 1. Indeed, equation 43 guaranteesthat α(x, t) ≤ α0(x) and, therefore, assumption 33 shows thatα(x,t) ≤ 1 for t ≥ 0,a.e. x ∈ Γ3. On the other hand, ifα(x,t0) = 0 at timet0, then it follows from 43 that.α (x, t) = 0 for all t ≥ t0 and therefore,α(x,t) = 0 for all t ≥ t0, a.e. x ∈ Γ3. Weconclude that 0≤ α(x, t) ≤ 1 for all t ∈ [0,T], a.e.x ∈ Γ3.

4. An existence and uniqueness result

Now, we propose our existence and uniqueness result.

THEOREM 2. Assume that 21-34 hold. Then there exists a unique solutionu,ϕ,β,α to problem PV. Moreover, the solution satisfies

(45) u ∈C1(0,T;V),

(46) ϕ ∈C(0,T;W),

(47) β ∈W1,2(0,T;L2(Ω))∩L2(0,T;H1(Ω)),

(48) α ∈W1,∞(0,T;L2(Γ3))∩Z.

The functionsu, ϕ, σ, D, β andα which satisfy 3-4 and 40-44 are called a weaksolution of the contact problem P. We conclude that, under the assumptions 21-34, themechanical problem 3-17 has a unique weak solution satisfying 45-48. The regularityof the weak solution is given by 45-48 and, in term of stresses,

(49) σ ∈C(0,T;H1),

(50) D ∈C(0,T;W ).


Indeed, it follows from 40 and 42 thatDiv σ(t)+ f0(t) = 0, div D = q0(t) for all t∈ [0,T] and therefore the regularity 45 and 46 ofu andϕ, combined with 21-29 implies49 and 50.

The proof of Theorem 2 is carried out in several steps that we prove in what fol-lows, everywhere in this section we suppose that assumptions of Theorem 2 hold, andwe consider thatC is a generic positive constant which depends onΩ,Γ1,Γ3, pν, pτ,γν,γτandL and may change from place to place. Letη ∈C(0,T;V) be given, in the first stepwe consider the following variational problem.

Problem PVη. Find a displacement fielduη : [0,T] →V such that

(51) (A ε(.uη (t)),ε(v))H +(η(t),v)V = (f(t),v)V ∀v ∈V, t ∈ [0,T] ,

(52) uη(0) = u0.

We have the following result for the problem.

LEMMA 1. There exists a unique solution to problem PVη which satisfies theregularity 45.

Proof. We define the operatorA : V →V such that

(53) (Au,v)V = (A ε(u),ε(v))H ∀u,v ∈V.

It follows from 53 and 21(a) that

(54) | Au−Av |V≤ LA | u−v |V ∀u,v ∈V,

which shows thatA : V →V is Lipschitz continuous. Now, by 53 and 21(b), we find

(55) (Au−Av,u−v)V ≥ mA | u−v |2V ∀u,v ∈V,

i.e., thatA : V →V is a strongly monotone operator onV. ThereforeA is invertible andits inverseA−1 is also strongly monotone Lipschitz continuous onV. Moreover usingRiesz Representation Theorem we may define an elementfη ∈C(0,T;V) by

(fη(t),v)V = (f(t),v)V − (η(t),v)V .

It follows now from classical result (see for example[2]) that there exists a uniquefunctionvη ∈C(0,T;V) which satisfies

(56) Avη(t) = fη(t).

Let uη : [0,T] →V be the function defined by

(57) uη(t) =

Z t

0vη(s) ds+u0 ∀t ∈ [0,T].


It follows from 53-57 thatuη is a solution of the variational problemPVη and it satisfiesthe regularity expressed in 45. This concludes the existence part of lemma 1. Theuniqueness of the solution follows from the uniqueness of the solution of the problem56.

In the second step, letη ∈C(0,T;V), we use the displacement fielduη obtainedin lemma 1 and we consider the following variational problem.

Problem QVη. Find the electric potential fieldϕη : [0,T] →W such that

(B∇ϕη(t),∇φ)H − (E ε(uη(t)),∇φ)H

(58) = (q(t),φ)W ∀φ ∈W, t ∈ (0,T) ,

we have the following result.

LEMMA 2. QVη has a unique solutionϕη which satisfies the regularity 46.

Proof. We define a bilinear form:b(., .) : W×W → R such that

(59) b(ϕ,φ) = (B∇ϕ,∇φ)H ∀ϕ,φ ∈W.

We use 20 and 24 to show that the bilinear formb is continuous, symmetricand coercive onW, moreover using Riesz Representation Theorem we may define anelementqη : [0,T] →W such that

(qη(t),φ)W = (q(t),φ)W +(E ε(uη(t)),∇φ)H ∀φ ∈W, t ∈ (0,T) .

We apply the Lax-Milgram Theorem to deduce that there existsa unique elementϕη(t) ∈W such that

(60) b(ϕη(t),φ) = (qη(t),φ)W ∀φ ∈W.

We conclude thatϕη(t) is a solution ofQVη , let t1,t2 ∈ [0,T], it follows from 20, 24and 58 that

| ϕη(t1)−ϕη(t2) |W≤C(| uη(t1)−uη(t2) |V + | q(t1)−q(t2) |W),

the previous inequality and the regularity ofuη andq imply thatϕη ∈C(0,T;W).

In the third step, we letθ ∈C(0,T;L2(Ω)) be given and consider the followingvariational problem for the damage field.

Problem PVθ. Find a damage fieldβθ : [0,T] → H1(Ω) such that

βθ(t) ∈ K, (.

βθ (t),ξ−βθ(t))L2(Ω) +a(βθ(t),ξ−βθ(t))

(61) ≥ (θ(t),ξ−βθ(t))L2(Ω) ∀ξ ∈ K a.e.t ∈ (0,T) ,

(62) βθ(0) = β0.

To solvePVθ, we recall the following standard result for parabolic variationalinequalities (see, e.g.,[19,p.47]).


THEOREM3. Let V⊂ H ⊂V′be a Gelfand triple. Let K be a nonempty closed,

and convex set of V. Assume that a(., .) : V ×V → R is a continuous and symmetricbilinear form such that for some constantsζ > 0 and c0,

a(v,v)+c0 | v |2H≥ ζ | v |2V ∀v∈V.

Then, for every u0 ∈ K and f ∈ L2(0,T;H), there exists a unique functionu ∈ H1(0,T;H)∩L2(0,T;V) such that u(0) = u0, u(t) ∈ K for all t ∈ [0,T], and foralmost all t∈ (0,T) ,

(.u (t),v−u(t))V′

×V +a(u(t),v−u(t))≥ ( f (t),v−u(t))H ∀v∈ K.

We apply this theorem to problemPVθ.

LEMMA 3. Problem PVθ has a unique solutionβθ such that

(63) βθ ∈W1,2(0,T;L2(Ω))∩L2(0,T;H1(Ω)).

Proof. The inclusion mapping of(H1(Ω), | . |H1(Ω)) into (L2(Ω), | . |L2(Ω)) is contin-

uous and its range is dense. We denote by(H1(Ω))′

the dual space ofH1(Ω) and,identifying the dual ofL2(Ω) with itself, we can write the Gelfand triple

H1(Ω) ⊂ L2(Ω) ⊂ (H1(Ω))′.

We use the notation(., .)(H1(Ω))′×H1(Ω) to represent the duality pairing between(H1(Ω))

′

andH1(Ω). We have

(β,ξ)(H1(Ω))′×H1(Ω) = (β,ξ)L2(Ω) ∀α ∈ L2(Ω),ξ ∈ H1(Ω),

and we not thatK is a closed convex set inH1(Ω). Then, using the definition 35 of thebilinear forma, and the fact thatβ0 ∈K in 34, it is easy to see that lemma 3 is a straightconsequence of Theorem 3.

In the fourth step, we use the displacement fielduη obtained in lemma 1 and weconsider the following initial-value problem.

Problem PVα. Find the adhesion fieldαη : [0,T] → L2(Γ3) such thatfor a.e.t ∈ (0,T)

(64).αη (t) = −

(αη(t)(γν(Rν(uην(t)))

2 + γτ | R τ(uητ(t)) |2 )− εa

)+

,

(65) αη(0) = α0.

We have the following result.

LEMMA 4. There exists a unique solutionαη ∈W1,∞(0,T;L2(Γ3))∩Z to prob-lem PVα.


Proof. For the sake of simplicity we suppress the dependence of various functions onΓ3, and note that the equalities and inequalities below are valid a.e. onΓ3. Considerthe mappingFη : [0,T]×L2(Γ3) → L2(Γ3) defined by

(66) Fη(t,α) = −(α(γν(Rν(uην(t)))

2 + γτ | R τ(uητ(t)) |2 )− εa

)+

,

for all t ∈ [0,T] andα ∈ L2(Γ3). It follows from the properties of the truncation op-eratorRν andRτ that Fη is Lipschitz continuous with respect to the second variable,uniformly in time. Moreover, for allα ∈ L2(Γ3), the mappingt → Fη(t,α) belongs toL∞(0,T;L2(Γ3)). Thus using a version of Cauchy-Lipschitz Theorem given in Theo-rem 1 we deduce that there exists a unique functionαη ∈W1,∞(0,T;L2(Γ3)) solutionto the problemPVα. Also, the arguments used in Remark 1 show that 0≤ αη(t) ≤ 1for all t ∈ [0,T], a.e. onΓ3. Therefore, from the definition of the setZ, we find thatαη ∈ Z, which concludes the proof of the lemma.

Finally as a consequence of these results and using the properties of the operatorG , the operatorE , the functionalj and the functionS, for t ∈ [0,T], we consider theelement

(67) Λ(η,θ)(t) = (Λ1(η,θ)(t),Λ2(η,θ)(t)) ∈V ×L2(Ω),

defined by the equalities

(Λ1(η,θ)(t),v)V = (G (ε(uη(t)),βθ(t)),ε(v))H +(E ∗∇ϕη(t),ε(v))H

(68) + j(αη(t),uη(t),v) ∀v ∈V, t ∈ [0,T] ,

(69) Λ2(η,θ)(t) = S(ε(uη(t)),βθ(t)), t ∈ [0,T] .

We have the following result.

LEMMA 5. For (η,θ) ∈ C(0,T;V × L2(Ω)), the functionΛ(η,θ) : [0,T] →V ×L2(Ω) is continuous, and there is a unique element(η∗,θ∗) ∈C(0,T;V ×L2(Ω))such thatΛ(η∗,θ∗) = (η∗,θ∗).

Proof. Let (η,θ) ∈C(0,T;V ×L2(Ω)), andt1,t2 ∈ [0,T]. Using 19, 22, 25, 26 and 27,the definition ofRν, Rτ and the remark 1, we have

| Λ1(η,θ)(t1)−Λ1( η,θ)(t2) |V

≤| G (ε(uη(t1)),βθ(t1))−G (ε(uη(t2)),βθ(t2)) |H

+ | E ∗∇ϕη(t1)− (E ∗∇ϕη(t2) |H

+C | pν(uην(t1))− pν(uην(t2)) |L2(Γ3)


Proof. Existence. Let(η∗,θ∗) ∈C(0,T;V ×L2(Ω)) be the fixed point ofΛ defined by67-69 and denote

(79) u∗ = uη∗ ,ϕ∗ = ϕη∗ ,β∗ = βθ∗ ,α∗ = αη∗ .

Let σ∗: [0,T] → H be defined by

(80) σ∗(t) = A ε(.u∗ (t))+G (ε(u∗(t)),β∗(t))+E ∗∇ϕ∗(t) ∀t ∈ [0,T] ,

and letD∗: [0,T] → H the function be defined by

(81) D∗(t) = −B∇ϕ∗(t)+E ε(u∗(t)) ∀t ∈ [0,T] .

We prove that the quadruplet(u∗,ϕ∗,β∗,α∗) satisfies 40-44 and the regularity 45-48.Indeed, we write 51 forη = η∗ and use 79 to find

(82) (A ε(.u∗ (t)),ε(v))H +(η∗(t),v)V = (f(t),v)V ∀v ∈V, t ∈ [0,T] ,

and we write 61 forθ = θ∗ and use 79 to obtain

β∗(t) ∈ K, (.

β∗ (t),ξ−β∗(t))L2(Ω) +a(β∗(t),ξ−β∗(t))

(83) ≥ (θ∗(t),ξ−β∗(t))L2(Ω) ∀ξ ∈ K a.e.t ∈ (0,T) .

EqualitiesΛ1(η∗,θ∗) = η∗ andΛ2(η∗,θ∗) = θ∗ combined with 68-69 show that

(η∗(t),v)V = (G (ε(u∗(t)),β∗(t)),ε(v))H +(E ∗∇ϕ∗(t),ε(v))H

(84) + j(α∗(t),u∗(t),v) ∀v ∈V,

(85) θ∗(t) = S(ε(u∗(t)),β∗(t)).

We now substitute 84 in 82 to obtain

(A ε(.u∗ (t)),ε(v))H +(G (ε(u∗(t)),β∗(t)),ε(v))H +(E ∗∇ϕ∗(t),ε(v))H

(86) + j(α∗(t),u∗(t),v) = (f(t),v)V ∀v ∈V,t ∈ (0,T) ,

and we substitute 85 in 83 to have

β∗(t) ∈ K for all t ∈ [0,T] , (.

β∗ (t),ξ−β∗(t))L2(Ω) +a(β∗(t),ξ−β∗(t))

(87) ≥ (S(ε(u∗(t)),β∗(t)),ξ−β∗(t))L2(Ω) ∀ξ ∈ K,


we write now 58 forη = η∗ and use 79 to see that

(88) (B∇ϕ∗(t),∇φ)H − (E ε(u∗(t)),∇φ)H = (q(t),φ)W ∀φ ∈W,t ∈ (0,T) ,

and we write 64 forη = η∗ and use 79 to find

(89).α∗ (t) =−

(α∗(t)(γν(Rν(u∗ν(t)))

2 + γτ | R τ(u∗τ(t)) |2 )− εa

)+

a.e.t ∈ (0,T) .

The relations 86, 87, 88 and 89 allow us to conclude that(u∗,ϕ∗,β∗,α∗) satisfies40-43. Next, 44 and the regularity 45-48 follow from Lemmas 1, 2, 3 and 4, since(u∗,ϕ∗) satisfies 45-46, it follows from 80 that

(90) σ∗ ∈C(0,T;H ).

We choosev = ω ∈ D(Ω)d in 86, we use 80 and 36 to obtain

Divσ∗(t) = −f0(t) ∀t ∈ [0,T] ,

whereD(Ω)d = u = (ui) / ui ∈ D(Ω) andD(Ω) is the space of infinitely differen-tiable real functions with a compact support inΩ, we use 28 and 90 to find

σ∗ ∈C(0,T;H1).

Let t1, t2 ∈ [0,T] , by using 24, 25, 18 and 81 we deduce that

| D∗(t1)−D∗(t2) |H≤C(| ϕ∗(t1)−ϕ∗(t2) |W + | u∗(t1)−u∗(t2) |V),

the previous inequality and the regularity ofu∗ andϕ∗ given by 45-46 imply

(91) D∗ ∈C(0,T;H).

We chooseφ ∈ D(Ω) in 88 and use 37 we find

divD∗(t) = q0(t) ∀t ∈ [0,T] ,

by 29 and 91 we obtainD∗ ∈C(0,T;W ).

Finally we conclude that the weak solution(u∗,σ∗,ϕ∗,D∗,β∗,α∗) of the piezoelec-tric contact problemP has the regularity 45-50, which concludes the existence part ofTheorem 2.

Uniqueness. The uniqueness of the solution is a consequenceof the uniquenessof the fixed point of the operatorΛ defined by 67-69.

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AMS Subject Classification: 74M15, 74R99, 74F99.

SELMANI LYNDA, BENSEBAA NADJET Department of Mathematics University of SETIF 19000SETIF, ALGERIAe-mail: [email protected]: bensebaa [email protected]


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