solution to signorini-like contact problems...

Nonlin.ar Ana/ysis. n.'Of)'. M.thods & Applications, Vol. 11. No. 12. pp. 1325-1350. 1987,Prinled in Grear Britain.

0362-546X,187 $3.00 + .00Petgamon Journals Ltd.

SOLUTION TO SIGNORINI-LIKE CONTACT PROBLEMS THROUGHINTERFACE MODELS-I. PRELIMINARIES AND FORMULATION OF

A VARIATIONAL EQUALITY

P. J. RABIERDepartment of Mathematics, University of Pillsburgh. Pittsburgh. PA 15260. U.S.A.

and

J. T. ODENTICOM. The University of Texas at Austin. Austin. Texas 78712, U.S.A.

(Received 7 February 1986: received for publica/ion 28 May ]986)

Key words and phrases: Interface models. contact-traction boundary conditions. variational equality.Sobolev spaces. nonavailabi1ity of Soholev's embedding theorems.

WE PRESENT a study of contact problems usually modeled by Signorini's problem. Our approachdiffers in that we make use of constitutive relations for the normal response along the candidatecontact surface. The form of these models is dictated by experimental evidence and they leadto a variational equality instead of an inequality. We have focused on the most delicate caseof contact-traction boundary conditions for which we obtain existence and optimal uniquenessresults under physically realistic assumptions. The other usual boundary conditions can bedealt with similarly with simplifications in the proofs. Signorini's problem is shown to berecovered as the limiting case of an infinite normal stiffness, while our model allows forperturbations describing friction phenomena, according to Coulomb's law or generalizationsof it.

Serious mathematical difficulties arise from the fact that the most general type of interfacemodels rules out the use of Sobolev's embedding theorem, without which the problem is nolonger in the province of standard convex analysis but rather lies in the realm of the theory ofhypermaximal monotone operators having a domain with empty interior. Several auxiliaryresults. including an apparently new property in Sobolev spaces. are proven which. togetherwith the general method of proof. should be of interest in other problems. The contact-tractionboundary conditions require compatibility conditions to be introduced. They have a somewhatmore sophisticated form than the standard ones involved in Signorini's problem and furtherexamination shows that their physical content agrees strikingly with common sense physicalobservations.

The work is divided into two principal parts: Part I. which focuses on preliminaries neededfor the analysis of the traction problem. special Sobolev space properties, and the formulationof the weak form of the nonlinear boundary-value problem; and Part II [1] which addressesquestions of existence and uniqueness of solutions,

1325

1320 P. J. RAB1ER and J. T. ODEN

I. INTRODUCTION

Let Q be an open. bounded subset of IRN with a Lipschitz continuous boundary r, thedisjoint union r cur F with r c and r F mcasurablc and mcas(rc) > O. For N = 2 or N = 3, Qrepresents the reference configuration of a body in geometrical contact with anothcr bodyalong r c. In other words, no cxternal forces are present and the boundarics of thc two bodiescoincide along r c. Supposc now that the body occupying the domain Q is submitted to externalforces r. consisting of body forccs b defined in Q and tractions t prescribed on r F' Assumingthat no point of r F may come in contact with the second body under the action of the forcesr, i.e. that r c is the candidate contact surface for the deformed configuration. and that thematerial has a linear elastic behavior, thc problem of contact with no friction is usuallyunderstood as the Signorini problcm: minimize

I2a(v, v) - (r. v)o,

over thc closed convex subset of the Sobolev space (H1(Q »N,K={VE(H1(Q»N,V,,~O on fd.

(1.1)

(1.2)

In (1.1) and (1,2) above, a( ... ) denotcs the symmetric bilinear form associated with the virtualwork of stresses u(u) on strains e(\')(a(u. v) = fo u(u): e(v) dx). ('. ')0 the duality pairingbetween the space (Ht(Q»N and its dual. and VII the component of the displaccment v alongthe outward normal vector n: v" = v . n (euclidian inner product). Problem (I. 1)-( 1.2) is aformal variational formulation of the equilibrium equations between the stresses u(u) and theexternal forces band t:

and. on fe,

div(T(u)+b=O inQ.

u(u)'n=t onfF.

u(u)'n=O if u,,<O,

(T(U) . n = -an, (l' ~ O. if Il" '* O.

(1.3)

(1.4)

(1.5)

(1.6)

(1.7)

A "qualification" of the formality of this interpretation can be found among the by-productsof our approach (d. remark 5.2).

Existence of solutions to problem (1.1 )-( 1.2) is known under a simple necessary and sufficientcompatibility condition on the applied forces [5. see also tOJ. The idea of using variationalinequalities for solving contact problems goes back to [5.10], but the theory has not been verysuccessful in the more complicated problem with friction despite recent contributions by Necas.Jarusek and Haslinger [6. 13].

With the aim of analyzing problems of friction. Oden and Martins [14] have developed adifferent approach to contact problems. The key ingredient of their theory is the introductionof a model for the normal response at points of r c at which contact may occur. To do this, itis essential to remove the nonpenetration condition Il" ~ 0 on r c. The normal response at apoint x E f c is then a function of (u,,)+(x). Contrary to a first natural reaction, removing thenonpenetration condition Il" ~ 0 is not physical nonsense. Indeed, in any mathematical model.

Signorini-like contact problems through interface models-I 1327

the boundary r c is an idealized averagc candidate contact surface. the real candidatc contactsurface differing from r c by a layer of asperities. How different thc real surface is from r c"measurcs" its roughness. a factor incrcasingly belicved to originate friction phenomena (see,e.g .. 114. 18,201). When contact occurs. the deformation of thcse asperities (incidentally of anature totally different from the "visible" deformation of the body) allows small displacementsof the boundary r c towards the obstacle, violating the condition II,,;:§; O. Accordingly. whenpositive, the displacement u,. should nevertheless rcmain small. This point will be examinedlater on. In this view. removing the condition ll,. ;:§; 0 docs not amount to accepting actualpenetration of the two bodies in contact but merely allows the average surface rc to get closerto the obstacle. On the experimental side, these features of actual surfaces have been observedby many investigators, and the memoir [14] contains extensive arguments in support of suchmodels. On the mathematical side. thcy present numerous advantages as follows. As we shallsee. discrepancies between variational and boundary value problem formulations vanish. ncwcompatibility conditions with precise physical interpretations are involved. perturbationsallowing for friction phenomena become manageable. etc.

In such models. the normal response caused by the normal displacement Il,,(x) at x Ere isthen of the form 1J(x. u,,(x») wherc 1J: rex ~ -+ [R verifies q, ~ 0 and q,(x. f) = 0 for f ;:§; 0 (sothat cp(x. Il,,(x») actually depends only on (Illl)+(x»), The function cp depends on the interfacecondition and a few of its propcrties are dictated by common sense observations: it is intuitivelyclear that any positive normal displaccment should produce a positive normal response. sothat q,(x. f) > 0 for f > O. Next. an increase of thc normal displacement must produce anincrease of thc normal response so that CP(x. t) is increasing w. r. 1. f ~ O. Further, the resistanceto penetration of thc bodics suggests for a positivc normal displaccment Il,,(x) that the ratioCP(x. Iln(~»)/ll,,(X) (normal response vs normal displacement) be an increasing function of lI,,(x).namely that q,(x. f)/f is increasing W.r.t. t > O.

Denoting by q,(11,,) the function ¢(II,J (x) = cp(x, llt.(X») for x Ere. the contact problem(1.3)-(1.7) becomes, in this approach:

find u E (H I(Q»)'" such that

div O'(u) + h = 0 in Q. (1.8)

O'(u) . n = t on rF, (1.9)

O'(u), n = -¢(un)n on rc. (1.10)

In Section 3, we shall see that problem (1.8)-(1.10) has the equivalent formulation: findu E (H1(Q)Y' such that

a(u,v)+J cP(ll")v,,ds=(f.v)Q forevery vE(H1(Q»)"'. (1.11)rc

Experimental evidence shows that the normal response CP(x. t) has a power-like behaviorfor small values of t > 0, growing up to exponential as t is increased. Roughly speaking. thepower zone ("light" normal loads) authorizes sliding. prohibited in the exponential zone("heavy" normal loads): see 114, Fig. 48]. For this reason, in a study of similar static contactproblems with friction (cf. Ili/), we have limited ourselves to considering the choice 1J(x, t) =c,,(x)(t+)'"n with (experimentally justified) values of the exponent mIl allowing the usc ofSobolev's embedding theorems and under convenient boundary conditions avoiding the needfor compatibility conditions.

1328 P. J. RAmER and J. T. ODIN

'.

This paper is devoted to the study of problem (1.8)-( 1.10) with a general ¢. By comparisonwith the situation in (11]. we face several new difficulties. First and foremost, the variationalformulation (1.11) is a priori not well posed since Sobolev embedding theorems arc notavailable without serious restrictions on thc growth of the function t- ¢(X./) as 1 tends to+ oc. In particular, exponential growth is prohibited when N ~ 3. This difficulty has beenovercome by requiring tP(II,,) to belong to the space O(f d as an additional condition to (1.8)-(1.10) and proving in this assumption that <p(lIn)vn belongs to L I(f d for every \' E (H l(Q»/lias soon as u is a solution to (1.8)-(1.10) (theorem 3.1). Once the variational formulation (1.11)has been justified, we show that it is equivalent to the minimization of a weakly sequentiallylower semicontinuous convex functional over (HI(Q»)N with values in IR' = lRU {+ oo}. In thisprocess. other difficulties arise because the functional in question is nowhere continuous inthe general case (i.e. its domain is empty), and the desired properties must be established byusing convexity of the function 1- n¢(x, r) dr rather than that of the functional. Anothertechnicality is to prove that the minimizers do verify equations (1.8)-(1.10) in the sense ofdistributions. and the condition 4>(un) E LI(fd.

From these introductory comments. one might have guessed that the problem is, in somerespects, pertaining to the theory of hypermaximal monotone operators (ct. [2] for an excellentaccount) rather than standard convex analysis. However, we have found no significant advan-tage in using the specialized vocabulary and the general results of this theory. while doing somight have caused some discomfort to the uninitiated reader. Nonetheless. it can be reasonablyspeculatcd on the basis of this relationship that our method of proof can be duplicated in otherproblems. thus ranging farther than the specific example for which it has been devcloped here.

Scveral general properties need to be establishcd. Some of them, specifically related tointegration theory, are collected in Section 2. In this respect. we note an interesting coincidcnce:most of the results of Section 2 hold undcr the apparently nccessary condition that themapping t- ¢(x. I) has (at most) exponential growth at infinity. Other statements of generalmathematical interest. rclated to a seemingly new property in Sobolev spaces. are proved inSection 3 (lemmas 3.3-3.6).

Coerciveness of the energy functional. hence existence of minimizers. is proved in Section4 [1] over an appropriate quotient space (although the functional is not quadratic!) and undera compatibility condition on the applied external forces slightly stronger than that needed forsolving problem (1.3 )-( 1.7). This compatibility condition is independent of the normal response¢ and. again, the necessary mathematical assumptions have significant physical counterparts.For instance, coerciveness for every external force f is obtained under a purely geometricalassumption bearing a striking interpretation. namely, that the body is "stuck" due to its contactalong f c in the reference configuration (in other words. contact along f c in the referenceconfiguration prevcnts the body from being moved without exerting external forces).

Uniqueness modulo the space .N' of infinitesimal rigid motions R vcrifying R" = 0 on f ('.hence uniqueness if .N' = {OJ. is obtained under another mild compatibility condition as soonas physical contllct occurs. In any case, uniqueness of the area of physical contact {x E f c;un(x) >O} is proved and it is shown that the same result is false in general if thc area of physicalcontact is rcplaced by the area of geometrical contact {x Ere; u,,(x) ~ O}.

Considering problem (1.8)-( 1.10) as a model for contact with no friction instead of Signorini'sproblem thus allows elimination of several ambiguities in the latter. An important question isthen to know how they relate to each other. In Section 5 [1J. we present a very simple answerproviding one more justification for the lise of normal response models: the Signorini problem

Signorini-like contact problems 1hrough interface models-I 1329

(2.1 )

coincides with Ihe case of an infinite normal response along fe. This takes us back to thenatural requircment that. when positivc, the displacement Un should bc "smal1" along f c.When u is a solution to problem (1.8)-(1. 10) and cp is a physically admissiblc normal response.a heuristic but strong argument in this direction is as follows. As a result of rcsistance topenetration, it is observed in physical expcriments that CP(x. t) is very largc for relatively smallvalues of t > 0 (with CP(x, t) = cn(x)(t + )mn. experiments have provided c,,(x) = cn in the rangeof 106_108

). On the other hand. by changing any normal response cp 10 )"cp and letting )"tend to + 00. we show in Section 5 that every solution u(),) is arbitrarily close to some solutionto Signorini's problem in the strong topology of (H'(Q»N. Hence. /I,.(),,) is close to zero in.say. every space U(f d such that ffl(r) <-. U(r). The interpretation of this result is that Uti isclose to zero on f c if the normal responsc CP(x, t) is large for relatively small values of t > O.which is precisely the actual physical situation.

Another advantage of the formulation (1.8)-( 1.10) is that it admits perturbations allowingconsideration of friction phenomena according to Coulomb's law or generalizations of it. Inthis case, it suffices to consider a power-like normal response (since the exponential zone ischaracteristic of no sliding). The method of [II] is thcn available with appropriate modifications(see [1. Section 5 D. The other aspects we discuss in Section 5 are the interpretations ofour compatibility conditions for coercivcnes.c; and uniqueness in the simple case N = 2, theadmissibility of an initial gap and the possibility of considering other boundary conditions. Theuse of interface modcls, such as those dcscribed here. together with nonquadratic energies ofdeformation. is the subject of current studies.

Thc work is naturally divided into two parts. Part I is composed of Sections 2 and 3. and isdevoted to the establishment of special mathematical preliminaries and to the formulation ofthe variational cquality which correctly characterizes the traction-contact problem with anonlinear interface constitutive law. Thc major issues of existence and uniqueness of solutionsare taken up in Part II [I].

2. TECHNICAL PRELIMINARIES

WC shall begin with a review of somc gcneral results. Hypotheses will later be complementedaccording to the applications we have in mind and further properties will be establishcd.

Let w bc an open subset of ~m, m ~ 1. and cp: w X ~ --+ ~. a Caratheodory function. i.e.

CP(x •. ): ~ --+ ~ is continuous for almost all x E (V,

CP( , . t) :Q --+ ~ is measurable for cvery t E ~.

The Nemytskii operator 1> associated with the function cp is defined for every measurablefunction ~: w --+ lR by

1>U;)(x) = cp(x. ;(x) for almost all x E (I). (2.2)

It can be shown that 1>([;) is a measurable function. Krasnoselskii [Y] has given necessary andsufficient conditions for the operator 1> to act continuously from U(w) into L'(w) when q andr verify the condition 1 ~ q, r < + 00. Besides. setting

~(x.~ =ft CP(x.r)dro

(2.3)

1330 P. J. RABIER and J. T. ODEN

and assuming q> 1 and r = q* = q/(q - I) (Holder conjugate of q). he has shown under thesame assumptions that the functional

j(;) = f cP(;) dx. (2.4)w

is continuously differentiable on U(w) with derivative 4>. in the sense that

j'(;)·h= f 4>(;)hdt.

'"(2.5)

for every pair (;, h) E (U(w)r. For q > 2. these results are complemented in [16} by showing,under the additional assumptions that (1) the mapping cP is continuously differentiable withrespect to (for almost all x E wand (2) its derivative CPt verifies an appropriate growth condition(ensuring that 4>tE ClI(U(w). U<//2)'(W»). that the functional j (2.5) is Iwice continuouslydifferentiable on U(w) with

f'(;) . (II. k) = f 4>t(;)ltk dew

for every triple (;,11, k) E (LQ(W»3. The same conclusion is false for q = 2. except in thetrivial case when CP(x. t) = a(x)t and (/ E L"'(w). but it can be extended to the case q = + 00,

of special interest to us in this paper. Proposition 2.1 below summarizes the various statementsfor q = + :x; in a form which will be suitable for our later purposes. Details and extensions canbe found in [17].

PROPOSITION 2.1. Assume that the Caratheodory function cP is of class C1 with respect to t foralmost all x E wt and that CPr verifies the following growth condition. For every to> O. thereis a function (/ttl EL' (w) such that

Icpl(·' 1)1 ~ (/to for III ~ to· (2.6)

Then

(i) 4>,E CO(L"'(w). L1(W»(ii) 4> E C1(L "'(w). L1(w» as soon as CP(', 0) E L1(w), with

DcP(;) ·It = cPt(;)h.

for every pair (;, h) E (C(w)J2.(iii) Setting

<l>(x. t) = r CP(x. r) dT.o

the functional

j(;) = J cP(;) dxw

t No additional assumptioll is needed to show that tP, is a Carathcodory function.

(2.7)

(2.8)

(2.9)

•

Signorini-likc contact problems lhrough inlerface modcls-I

is twice continuously differcntiable on LX( (I) and

j' (;) . II = f 4>( ;)h dx,,,'

for every pair (;, II) E (L "'(w»)2 (note that the combination of (i) and (ii) yields

j"(;) . (II, k) = f 4>t(;)hk dx0'

for cvery triple (;. h. k) E (L"'(w»)3).

Suppose now that

<1>( . , f) ~ 0 for every f E IR.

1331

(2.10)

(2.11)

(2.12)

Then, for evcry measurable function ~: w --" IR, (~) is a nonnegative measurable function.Thc functional j of (2.9) can thus be cxtcnded to all mcasurable real-valucd functions as amapping (stiIl dcnoted by j) with values in ar = IRU {+ cx:} by setting

. -{f (g)elx if (;)EV(w))(;) - to

+ x othcrwise.

PROPOSITION 2.2. The extended functional j is lower semicontinuous on L I( w).

(2.13)

(2.14)

(2.15)

Proof. The result follows by combining [4, p. 218. proposition 1.1 and p. 222, corollary1.2J. •

Remark 2.1. The same argument shows that the extended functional j is lower semicontinuouson U(co) for every I ~p ~ + x.

We shall considcr a particular case when assumption (2,12) is satisfied, namely

q,,(-,f)iii;O for f>O.

q,t( .. t) ; 0 for t ~ 0

and

q,(x, f) ; r q,r(X, r) elr (¢:> q,( .. 0) ; 0).[)

(2.16)

In this case, the three functions q,t( .. f). q,( .. t) and <1>( . , t) are nonnegative for every t E lRand vanish for ( ~ n. Together with an appropriate condition limiting the growth of the functionq,(x, t) as t tends to + =>0. the above assumptions will now allow us to complement proposition2.1 as follows.

PROPOSITION 2.3. Assumc that (2.15) and (2.16) hold and suppose further that the functionq,t(x .. ) is nondecreasing for almost all x E wand that there are constants T> 0 and II> 0such that

cjJ,('. f) ~ ,ucjJ(·. t) for / iii; T. (2.17)

1332 P. J. RAB1ER and J. T. OOEN

....

with 4>('. T) (=4>(T)) E Li( w) and either 4>(x, T) > 0 or 4>(x, . ) ~ 0 for almost all x E w,Then. for every measurable function S such that 4>(s) E L I(W) and cvcry function 1/ E L"'(w)one has

and the functional

1/ E L" (w) -+ h (1/) = j( s + 1/)

is real-valued and twice continuously differentiablc with

jH'1) . II = J 4>(s + 1/)11 dx,w

i'i(1/)' (h, k) = J 4>t(S + 1/)hkdx.w

for every triple (1/. h. k) E (L "'(w»)3.

(2.18)

(2.19)

(2.20)

(2.21 )

Proof. As a first step, we show that 4>(s) belongs to Li(w) as follows. Integrating both sidesof inequality (2.17) we obtain

4>(- , t) ~ 4>(- , T) + Wl>(- • t) - W'P(- • T)

~ 4>( . , T) + µ<l>( . ,t) for t ~ T.

Let then s:w -+ lR be a measurable function. From (2.21),

;(x) ~ T~ 4>(x, ;(x» ~ rjJ(x, T) + µ(x. ;(x».

On the other hand, it follows from (2.15) that the function rjJ(x, . ) is nondecrcasing for almostall x E w. Hencc.

;(x) ~ T~ 4>(x, ;(x» ~ 4>(x, T) ~ cjJ(x, T) + µ(x, ;(x).

This shows that

o ~ 1>(;) ~ <1>(T) + (1<1>(;) E L1(w),

proving the relation <1>(;) E L 1(w).Setting

4>;(x, t) = 4>(x, Sex) + t). (2.22)

we now note that 4>~is a Caratheodory function, and the mapping t -+ 4>;(x, t) is continuouslydifferentiable for almost all x E w with

(4);)t (x. t) = q>t(x, ;(x) + t). (2.23)

The sole nontrivial part of this assertion is the measurability of the function 4>;(" t) forevery t E IR. However, it suffices to notice that the function (x, T) E w x ~ -+ 4>(x, T + t) is(obviously) a Caratheodory function and to apply the mcasurability rcsult used at the beginningof this section after rcplacing T by ;(x).

...,


At this stage. we see that the properties to be established follow from proposition 2.1.provided we can prove that the function <P~ fulfills the required hypothcses. As <P;(. , 0) =<P('. ~(.» = cP(;) is in L lew). as we have just seen, we need only prove for every (0) 0 thatthere is a function b 10 E V (w) such that

o ~ (<P;)t(x, () ~ bto(x) for almost all x E wand It I ~ to. (2.24)

We begin by observing that an equivalcnt formulation of assumption (2.17) is

<p(·.r+()~el't<p(·.T) for (~O and r~T.

the proof of which reduces to a simple vcrification. In particular.

<P(. , t) ~ el'(t- T) <P(. , T) for (~T.

Substituting into (2.17). we arrive at

tpt( . , t) ~ µ el'(t- T) tp( ., T) = IJ ell(/- n 4>(T) for t ~ T.

Meanwhile. from the monotonicity of almost all functions tpt(x .. ). one has

<p'(-. () ~ tp,(-, T) ~ Iltp(-, T) = IJ4>(T) for (~T.

Hence, given to> 0 and setting

ato = IISUp(1, el.{to-n) 4>(T) E V(w).

we obtain the estimate

O~<Pt(·.()~ato for Itl~to.

(2.25)

(2.26)

(2.27)

To prove (2.24), we may assume (0 ~ T. since the same function bTcan be taken as bto whento ~ T. We shall find an appropriate choice for b,o by considering the three cases: ';(x) ~T - (0;

T - to < ;(x) < T; and ~(x) ~ T. Assume first ';(x) ~ T - to. Then. for lei ~ to. one has~(x) + t ~ T. From (2.23) and (2.26), we deduce

o ~ (tp;),(x, t) ~ µ4>(T)(x). (2.28)

Next. suppose T - to < ;(x) < T. Since to ~ T> 0 by hypothesis and for lei ~ to. we see that-2to < ~(x) + l < 2lo. Applying (2.27). we get

o ~ (<P;)t(x. t) ~ a2to(x). (2.29)

Finally, assumc that ;(x) ~ T. For 0 ~ t ~ to, relation (2.25) is available with r = ~(x) and weobtain

More generally. as soon as ';(x) + t ~ T. rclations (2.17) and (2.23) yield

o ~ (tp~)t(x, t) ~ Iltp(X, ;(x) + t).

Together with the previous inequality. we find

o ~ (<P;),(x. t) ~ Ilel'to cP(;)(x). (2.30)

1334 P. J. RAtHER and J. T. ODEN

for 0 ~ 1 ~ 10' Next. due to the monotonicity of almost all functions <P(x, . ) and from (2.17):

o ~ (<PE)I(X. t) ~ µ<P(x. ;(x)) = 11IP(;)(X)

when t < 0 and ;(x) + t ~ T. This shows that inequality (2.30) is valid whenever ;(x) ~ T.ld ~to and ;(x) + t ~ T. It remains to examine thc case when ;(x) ~ T and ;(x) + 1< T. If so, I isnegative and

-/0 < T-/o ~ ;(x) + 1< T~ (0,

whence, from (2.23) and (2.27),

(2.31)

According to the estimates (2.28)-(2.31) and since <P(T), <P(;), ato and a2J" belong to thespace LI(w), we can take

blu = SUP(ll<P(T), !lelllU <p(:;). alo' a2to) E LI(U)

in (2.24) and the proof is complete. •

By localization and partition of unity, the rcsults of this section easily carryover to the casein which the open set w is replaced by the Lipschitz continuous boundary r of a (bounded)open subset Q of jRt,'. Indeed, this merely introduces positive measurable bounded weights,which does not affect the form of the required hypotheses. Further. r can also be replaced byany measurable subset rc for it is immediately secn that the assumptions arc not affected byextending all thc data by zero for values of the variable x in nrc, Theorem 2.1 belowsummarizes the conclusions in this new context.

THEOREto.l2.1. Let Q be an opcn bounded subset of jRN with Lipschitz continuous boundary rand surface measure ds and let r c C r be a measurable subset. Let <p: rex R -+ iR be aCaTathcodory function and assume for almost all x E rc that the function <P(x •. ) is continuouslydifferentiable with <Pt(x .. ) nondecreasing. Assumc furthcr

<Pt( .. I) ~ 0 for (> O.

<P,( .. I) = 0 for I ~ O.

cp(x,t)= r <Pt(x.T)dT(~<P(·,O)=O).o

(2.32)

(2.33 )

(2.34)

and that there are constants T> 0 and 11 > 0 such that

<Pt(', () ~ !l<Pt(·. t) for I ~ T, (2.35)

with <P( .• n E L f(rd and either <P(x. T) > 0 or lp(x .. ) ;3 0 for almost all x Ere. Setting

(x,t) = r <P(x. T) dT (2.36)o

and denoting by cPt. cP and the Ncmytskii operators associated with <P" rcspectively,the following conclusions hold:

,

(i) the functional

Signorini-likc contact problems through interface models-I

. {f . <b(~) ds if <b(~) ELI (f clJ(;) == r c

+ x otherwise,

1335

(2.37)

defined for every measurable function;: fc-+ IRis lower semicontinuous on the space L l(r c);(ii) for every measurable function; such that <beg) E L l(f(') and every function 71E LOC(r d

one has

<PCs + 11)E V (rd. <Pt(; + 71) ELI (re>and the functional

is real-valued and twice continuously differentiable with

h(q) . h == f <P(g + TI)h d.\'.rc

jg(11) . (h, k) == f <P,(s + 11)hk d~.rc

for every triple (1/, h, k) E (L"'(f c»)3.

(2.38)

(2.39)

(2.40)

(2.41)

Remark 2.2. Functions verifying the assumptions of theorem 2.1 generate a convex conecontaining all functions of the form

cp(x, I) = c(x)(l+)"'.

where m> 1 is an arbitrary real number, 1+ is the function

-t if t ~ O.l+(t)={ o if t < O.

and eEL '(f c) is an arbitrary nonnegative givcn function. Moreover. condition (2.35) allowsfor a modification of the growth of (convex combinations of) such functions as t tends to + 00,

up to and including exponcntial order.

The assumptions on the function cp madc in theorem 2.1, and in particular limitation of thegrown in I to exponential order. will later allow us to characterizc the solutions to the problemconsidered in Section 1 as minimizers of a convex functional with values in lR'. As usuaLexistence of minimizers will be obtained by cstablishing a coerciveness property, relying oncomlementary (and. of coursc. compatible) specification of thc growth of thc mapping cp(x.. )for almost all x E rc. The suitable hypothesis and its main consequences regarding the problcmunder consideration arc examined as the next objective of this scction.

As an additional assumption. we shall require that

cp(x, I)o ~ - ~ CPt(x. t) for t > 0 and almost all x E r c.I

-

1336 P. 1. RABlER and 1. T. ODEN

(2.42)

Note since the function lj>(x, . ) is continuously differentiable that this new assumption implieslj>(' ,0) = 0 and hence makes condition (2.34) superfluous. Besides, it is easily checked that(2.41) amounts to saying that the function t-lj>(x, t)/t is nondecreasing on (0. + 00) for almostall x E [c. This function is strictly increasing on (0, + 00) under the mildly stronger condition

lj>(x, t)0<- < lj>t(x, t) for t > O.

t

Monotonicity and nonnegativity properties allow us to set for almost all x E [c

. lj>(x. t) .{(x) = 11m- ~ 0 (possibly + 00)

t->+x t (2.43)

(2.45)

and l(x) is equivalently defined through any sequence (lj>(x, tt)/tt) with lim tt = + 00 and verifies

lj>(x. t){(x) ~ - for every t >0 and almost all x E [c. (2.44)

t

As a result of Egorov's theorem, I is a measurable function on [c.

Remark 2.3. It is obvious that functions verifying property (2.41) form a convex cone and thatthis assumption does not restrict the class described in remark 2.2. This shows that addingcondition (2.41) is compatible with the assumptions of theorem 2.1, which is due to the factthat T> 0 can be taken arbitrarily large in (2.35). Roughly speaking. the combination of (2.35)and (2.41) means that the growth of lj>(x, .) is superlillear and at mosl of exponential order,monotonicity being imposed by (2.32).

THEOREM2.2. In addition to the hypotheses of theorem 2.1. t assume that (2.41) holds and lets:[c- [R be a measurable function. Thcn, the mapping

11-"", j(IS) E [R',t-

where j denotes the functional (2.37). is nondecrcasing on (0, + 00) and

I 1 flim "'"j(ts) = - In ds:j:/->+'" 1- 2 rc

(2.46)

where S+ = sup(g, 0) and with the (usual) convention that

I(x);~(x) = 0 when l(x) = + 00 and ; ...(x) = O.

Proof. A preliminary observation is that for almost all x E [c. the function 1- (x, t)/12 isnondccrcasing on (0, + 0;;): To see this, it suffices to compute

a ((X, I») 1- ~ ="3 (1lj>(X. I) - 2<1>(x, I»al 1- I

t Although assumption (2.35) can be omitted in this statement.

:j: Allowing, of course, the value +:x: for the right-hand side.

,


and note that Il/>(X, I) - 2«P(x. I) ~ 0 for I> 0 as it follows by multiplying (2.41) by I andintegrating. This yields the monotonicity of the function (2.45) from the relation (for t> 0)

1 {«P( x, t;(x»(2 4>(x, t;(x» = 12~2(X) ;2(x) for ;(x) > O.

o for ;(x) ~ O.(2.47)

A second conclusion from the above observation is that lim 4>(x, t)/t2 exists (possibly + (0)I~+:I;

for almost all x E rc. Actually, a more precise result is true, namely.

lim «P(~. I) =! lim l/>(x, I) = ~ I(x)./-++00 t 2t-++" t 2

(2.48)

Although it holds in a much more general contex, this relation is easy to prove under ourassumptions. Indeed. it is a simple exercise to check that (2.48) follows from de l'H6pital'srule.

We are now in position to prove relation (2.46) as follows. Suppose first that the left-handside of (2.46) is a (nonnegative) real number 1. For every sequence tk tending to + x. one has

and

1 f -(2 «P(lk;) ds .:;;/ for every k E ~.k rc

(2.49)

(2.50)

On the other hand. it follows from (2.47) and (2.48) (and the convention I(x);~ (x) = 0 whenI(x) = + 00 and ;+(x) = 0) that the nondecreasing sequence 4>(x. ik ;(x»/t~ tends tol(x)n(x)/2 almost everywhere on rc. With (2.49), (2.50) and the monotone convergencetheorem. it follows that

~f In ds = I.rc

To complete the proof. we shall use the inequality

4>(x, t) s; ~ lex) for (> 0,(2 - 2

which follows from (2.48) and the mono tonicity of «P(x, t)/12. With (2.47). this yields

1, 1_2 /;+ ~ (2 «P(I;) for (> O. (2.51)

1338 P. J. RAuum and J. T. ODEN

-.

Assume next that the right-hand side of (2.46) is a (nonnegative) real number. namelyl;i ELI (f C). If so. relation (2.46) follows from Lebcsgue's dominated convergcnce theoremby using a sequence (k > ()tending to +:x; in (2.51). As both sides of (2.46) coincide wheneither one is finite. they coincide when either one is + 00 as well. and the proof is complete. •

3. VARIATIONAL FORMULATION

This section is intended to show the equivalcnce of the contact problem described in Section1 with a minimization problem over the space (HI(Q »,\'. We shall bcgin with a revicw of someclassical results and introduce a few notations to be used throughout the remainder of thispaper.

Givcn a boundcd domain Q with a Lipschitz continuolls boundary f. the outer normal 11 isdefined almost everywhere (sec. e.g., [12]) and is a measurable function. Componentwise, wethen have

1/, E L X(f). I ~ i ~ N.N

~ tl1 = 1 on f.;=1

(3.1 )

The space H1(Q) is the usual Sobolev space of distributions with partial dcrivatives oforder ~1 in L2(~2). The space (H1(Q »)N is endowed with the usual inner product inducingthe norm

(3.2)

The trace operator maps the space (HI(Q»)'''' linearly and continuously onto the space(H1/2(f))N. with topological dual

(3.3)

From (3.1) and for every 1 ~ P ~ + ::c, a given element g E (U(f»)N has a decompositionof the form

with

g = gT + ;/1"'

;n = g . n = ;;Ili E U(f),

gT = g - ;nl1 E (U(f»)N

(3.4 )

(3.5)

The components ;tl and gT will be referred to as the normal and tangential components of grespectively.

Signorini·like contact prohlems through interface models-I 1339

We next make precise the assumptions on the data of the problem as follows. We shallassume that the elasticity coefficients Eijk/ verify

Eijki E L"'(Q)

and that the uniform ellipticity condition

Eijk/(x)Ak/Aij ~ aAijAij

holds with some constant a for almost all x E Q and every N x N symmetric array Aij.

The body forces are chosen so that

(3.6)

(3.7)

while the prescribed tractions t on f F = nf c are submitted to the condition

t E (U(f F)t. (3.8)

(3.9)for r > Ot

The function ¢ :rc x IR-+ IR characterizing the normal response along rc is supposed tofulfill the assumptions of theorem 2.2. namely ¢ is a Caratheodory function such that themapping ¢(x .. ) is continuously differentiable for almost all x E rc and

o ~ ¢(- , r) ~ ¢t (- . r)r

¢,(x, . ) is nondecreasing for almost all x E rc.

</>,(- • r) = 0 forr~O,

(3.10)

(3.11)

and there arc constants T> 0 and µ> 0 such that

</>,(- . r) ~ IUP(- , t) for t ~ T. (3.12)

with ¢( .. T) E L l(fd and either ¢(x, T) > 0 or ¢(x, . ) = 0 for almost all x E f c.

Physical justification of thesc assumptions is provided by rcmarks 2.2 and 2.3 and the relatedcomments of Section 1.

In what follows. we shall denote by a( . , .) the continuous bilinear form over (H1(Q)Ndefined by

(3.13)

a( u. v) = virtual work produced by the action of the stresses

u(u) on the strains E(l')

= f u(u): E(V) dx = f Eijkla/UkajVi dx.Q Q

The external forces (body forces b and prescribed tractions t) are described by an elementf E [(HI(Q»N}' through the relation

(f, V)~2 = f b' v dr + J t· vcls. (3.14)Q rF

t Recall thaI this condition implicitly contains (2.34).

1340 P. J. RABIER and J. T. ODEN

a formula in which {'. ')0 stands for the duality pairing between (HI(Q»N and its topologicaldual.

Our aim is to examine how the problem (1.8)-(1.10) relates to the minimization problem:minimize

1 f-za(v.v) + ¢(vlI)cIs-{f,v)orc

(3.15)

where, as in Section 2, is the Nemytskii operator associated with the function (x. t) =Jb cp(x, r) dr. However. as the integral f rc cP(vlI) cis is not defined in general. the correctformulation of the above problem is: minimize

where, for every measurable function;: f c-lR, we have set

. {f cP(;) cis if <iJ(;) E V (f C).J(;) = rc

+00 otherwise.

(3.16)

The first step of our approach is classical and relies upon a generalization of Green's formula,given in lemma 3.1 below, whose proof can. for instance, be found in [7. theorems 5.8 and5.9]. We denote by ®(Q) the space of indefinitely differentiable functions with compactsupport in Q, equipped with the usual inductive limit topology, and by ®'(Q) its topologicaldual. the space of distributions over Q. We shall also make use of the spaces ®(Q) and ~(f)of restriction to Q and r, respectively. of indefinitely differentiable functions on IRN.

Consider the operator

div u: (HI(Q»N _ (~'(Q»N

{div u(v)]; = djGij(V) = dj(Eijk/d,Vk)

and define the subspace Hdivu(Q) of (HI(Q»N by

Hdivu(Q) = {v E (HI (Q »"'; div u(v) E (U(Q »N}.

(3.17)

(3.18)

LEMMA 3.1. (generalized Green's formula). There is a unique linear continuous operator

1r: Hdivu(Q)- (H-t/2(Q»N.

satisfying{1r(U)]; = G;j(U)lIj on f

for every;) E (Ht(Q»N such that Gi;CU) == Ei;ktO/Uk E <€1(Q), 1 ~ i. j. k, I ~ Nand

a(u,v)+ f (divu(u»·vdx={1r(u),v)r. (3.19)Q

for every u E Hdiva(Q) and every v E (H1(Q»N, where (. , ')r denotes the duality pairingbetween (HI/2(Q »N and (H-l/2(Q »N.

,.. Signorini-likc contact problcms through interface models-I

Due to the above lemma. a generalized form of problem (1.8)-(1.10) is as follows.Find UE (H1(Q»)N such that

1341

with the boundary conditions

divu(u)+b==O inQ. (3.20)

Jf(lI) == t on f F,

Jf(u) ==- cP(u,,)n on f c.

(3.21)

(3.22)

Note, however. that this formulation contains some ambiguity. Indeed, Jf(u) is merely in thespace (H-J/2(Q »)N while ¢(u,,)-hence, ¢(un)n-is only a measurable function. As Jf(u) is nota function in general, and a measurable function does not induce a distribution in a canonicalsense, it is not clear how relation (3.22) must be understood. To circumvent this difficulty. weshaIl include. as a part of the problem, the condition

(3.23)

Such a condition does make sense in (~'(f))N: it means that the action of the distribution Jf(u)on elements g E (~(f»)N is represented (in a necessarily unique way) by some element of(L1(f))N. As usual. the notation Jf(u) stands for both the distribution and its representativein (LJ(f»)N. Conditions (3.21) and (3.22) can then be understood in the sense of measurablefunctions, namely. almost everywhere. We now give a first characterization of solutions toproblem (3.20)-(3.23).

LEMMA 3.2. An element U E (H1(Q»N is a solution to problem (3.20)-(3.23) if and only if¢(utl) E Ll(fc) and

a(u, v) + f ¢(u")v,,ds==(f,v)o forevcry vE(~(Q»)'\'. (3.24)rc

Proof. Let u E (H1(Q»N be a solution to problem (3.20)-(3.23). From (3.22) and (3,23),one has d>(un)n E (L I(f d)N and hence CP(un) == cp(un)n . n E V(f d. Multiplying (3.20) byvE (0J(Q»)N and integrating over Q, the generalized Green's formula (J.19) of lemma 3.1yields

0== I (div U(lI) + b) . v dx == (Jf(lI). V)r - a(lI, v) + I b· v dx. (3,25)o 0

The expression (Jf( u). v)r depends on the restriction of v to the boundary f only, an elementof (~(f))N. Thus. applying (3.23). we get

(Jf(u), v)r == f Jf(u)· v ds. (3.26)r

Using a decomposition of the integral over f into integrals over f c and f F and from (3.21)and (3.22), (3.26) reads

(n(II). v)r == ft. vds + f ¢(un)n' vds.rF rc

(3.27)

1342 P. J. RABIER and J. T. OOLN

As ¢(un)n . \. = cP(II,,)Vn by definition of LJ". it suffkes to combine (3.25) and (3.27) and usethe definition (3.14) of the external forces f to see that u is a solution to equation (3.24).

Conversely. let u E (Hl(~~)Y\' be a solution to equation (3.24) such that 4>(11,,) E L1(fc).Taking v arbitrary in (5l(Q»'" and by definition of f (d. (3.14» it is immediate that div (T(U) =b in (~'(Q»N and hence U belongs to the spacc Hdi\(T (3.18). Together with thc gencralizedGreen's formula (3.19) of lemma 3.1, we see for an arbitrary \' E (~(Q»'" that

f cP(II")V,, dl' + (n(u). ,.)1' = ft. v ds,rc rF

or. equivalently.

(.iT(u). vh· = - f cP(II")V,, ds + ft. V d\'.rc rF

This relation involves restrictions of clemcnts of (51(Q»N to the boundary f only. and hcncecan be equivalently stated for an arbitrary" E (51(n )N. As cP(II,,) E V (l"d. one has4>(u,,)n E (Ll(fC>Y" and it follows that the distribution 1I'(u) is represented by the element of(V(n)'" defined by t on fF and by 4>(II,,)n on fe. Hencc. u is a solution to problem (3.20)-(3.23), •

Let u E (H1(Q».\' be a solution to problem (3.20)-(3.23). From relation (3.24). it is clearthat the mapping v E (51(Q»)'" - Ire cP(u")v,, d\' extends as a lincar continuous form over(H1(Q ) )"v. say (cP(lIt,)n. v)g (depending on thc trace of \' on f only) and relation (3.24) rcmainsvalid with (4)(II,,)n. v)o replacing Ire cP(II")V,, ds f~r an arbitrary v E (HI(Q )Y". On the othcrhand, from theorems 2.1 and 2.2, the cxpression (fj>(II,,)n. \')0 strongly resembles the dcrivativeat u of the functional

(3.28)

where j is the functional (3.16). Strictly speaking. this is not true since the functional (3.28) isnot differentiable (for instance. because it may take thc value + (0). In some cases. lack ofdifferentiability for convex functionals is not too serious a problem, but a standard assumptionis that the functional at least have a domain with nonempty interior. i.e. be continuous on anonempty open subset. Here. this condition is not satisfied in general: for choices of Nand fj>such that the Sobolev embcdding theorems are not available. the functional (3.28) mayperfectly be continuous at no point of (HI(Q »'" since it may take the value +x near any pointv with j(LJ,,) < +00. The idea of using Orlicz-Iike spaces is complicated by thc fact that thefunction cp is allowed to depend on the point x E fc. Besides, it is not clear at all that changingthe space (HI(Q»,\' into a smallcr one. ovcr which the functional (3.28) would have nicerproperties, would not lead to later problems (as far as coerciveness is concerned for instance).Despite the fact that the functional (3.28) is convex, we have then no standard way of justifyingthe arguments with which it is formally easy to deduce that solutions to problem (3.20)-(3.23)are minimizers of the functional (3.15).

To make up for the lack of regularity of the functional (3.28). we shall adopt an approachbased on convexity properties of the function 1> instead of convexity of the functional (3.28)directly. But. to do this. it is essential to obtain further information on the terlll (4)(un)n, v)ofor a general v E (HI(Q »N: The next few results are devoted to proving (in lemma 3.6) for

Signorini-likc contact problems through interface modcls-[ [343

every v E (H'(Q»N and provided that u is a solution to (3.20)-(3.23) that cP(l/n)Vt• E L t(f dand (<f>(lln)n. \')Q = Irc¢(Il")v,, d~. cxactly as whcn v E (9J(Q ». A somewhat surprisingassertion in which non negativity of the function <f> is one of the two keys.

LEMMA 3.3. Let I' ~ 0 be an element of L I(fd and suppose that the mapping

v E 9J(Q) - f Tv d\' (3.29)rc

extends as a linear continuous form (I'. v)Q over the space H1(Q). Then. for every v E f[1(Q).onc has Tv E LI(fcJ and

(I'. v)Q = f Tv ds for every v E HI (Q). (3.30)rc

Proof. Let Il E }-[l(Q) be given. Suppose tlrst 0 ~ v ~ M for some constant AI. Using theclassical procedure of e~tensiol1 to Hl(fRN) and regularization. it is easy to find a sequence(V(k» of elements of 9J(Q) tending to 0 in fjl(Q) and verifying 0 ~ V(k) ~ M. After extractinga sub-sequence. we may assume that vtk) tends to v almost everywhere on f c. From thecontinuity of (I', ')\1 on the one hand and Lebesgue's dominated convergence theorem on theothcr hand. we get (I'. v)o = lim Ire Tv cis and Tv E U (fe) with Ire Tv ds = lim I rc Tv(k)ds. Thesc relations prove (3.30) when 0 ~ v ~ M.

Suppose next that v ~ O. For every kEN - {O}set

/J(k) = inf(v. k).

Clearly. 0 ~ Vlk) ~ k and Vlk) E H'(Q) for evcry k (cf. [12, lemma 1.1. p. 313]). Arguing as in[12]. it is easily seen that Ilv(k%.Q ~ IIvlll.o, As V(k) tends to v in U(Q). it follows that v is theunique cluster point of the sequence (/P» in the weak topology of H1(Q ) and. hence. vlk) ~ vin f[l(Q). As a result. (I', v)Q = lim(T. v(k)h But (I'. V(k)O = f fc Tv(k) tis from the first partof the proof. so that

(I'. v)Q = lim Ire Tv(k) ds. (3.31)

(3.32)

From the non negativity of 1', the sequcnce (Tu(k» is nonnegative and nondecrcasing and tendsto Tv almost everywhere. Applying the monotone convergence thcorem, we find

f Tv ds = lim f Tv(k) cis.rc rc

The combination of (3.31) and (3.32) shows that Tv E L '([ c) and (3.30) holds when v ~ O.Finally. let I' bc arbitrary in H1(Q) and write v = "+ - v_ with v+ = sup(v. 0). v_ =

- inf(v. 0). From [12. lemma 1.1. p. 313]. we know that v+ and v_ belong to H1(Q). Thus.

1344 P. J. RABlER and J. T. ODEN

From the above and since V+ and v_ are nonnegative, 1'v+ and 1'v_ are in L I(fc) and thisrelation reads

(1',v)Q= f Tv+ds- f Tv_ds= f Tvds.fc rc rc

which completes the proof. •

The following result is a first extension of lemma 3.3 to vector-valued functions.

LEMMA 3.4. Let T E (L'(f c»N be given and suppose that the components Ti of T in thecanonical basis of lRN verify Tj ~ 0, 1 ~ i ~ N. Suppose also that the mapping

v E (qjJ(Q»N ~ fT' v dsfc

extends as a linear continuous form (T, v)Q over the space (Hl(Q»N. Then. for everyv E (Ht(Q »N. one has T . vEL '(f c) and

(T, v)Q = J T· v ds.rc

Proof. Let 1 ~ i ~ N be fixed and take v E H'(Q). Denote by v E (H1(Q »N the vector-valued function whose comEonents of order j *' i are 0 and whose ith component is v. Forv E 9l(Q), one has v E (9l(Q»N and

(T,v)o=J T.Vds==J Tivds.rc rc

As the mapping v E HI(Q)- v E (HI(Q»N is obviously continuous. this shows that themappmg

v E 9l(Q)~ J Tiv dsrc

extends as a linear continuous form (Ti, v)Q over H'(Q). Applying lemma 3.3. it follows thatTiv E Ll(fd for every v E H1(Q) and

(Ti• v) = f Ttv ds.rc

From the denseness of (9l(Q»N in (H1(Q »N, the identity,AI

(T, v)Q == L (1'i• vjh~j=1

for v = (Vj) E (9l(Q»N remains valid for v == (v;) E (H1(Q »N and thus readsN

(T. v)a ==L J Tjlli ds ==J T· v ds. •t=l rc rc

Lemma 3.4 can be considerably generalized as follows.

Signorini·like contac1 problems through interface models-I

LEMMA 3.5. LetT E (Ll(f c))N be given and suppose that the mapping

vE(211(Q)y-f T·vdsrc

1345

extends as a linear continuous form ('I', v)Q over the space (H1(Q))N. Let (fk) be a finitecovering of f by open subsets and suppose for every k that there is a system of coordinates inIRNin which Tj ~ 0 on fen fkt, 1 ~ i~N. Then. for every v E (H1(Q »)N one has T . v E L'(f dand

(T.v)a= f T·vds.rc

Proof Each open subset fk of f is the intersection Uk n f of f with an open subset Ukof Q. Let (B, Bk) be a partition of unity associated with the covering (Q. Uk) of Q. Forv E (211(Q»)N, one has Ehv E (211(Q»)""and

f T'lhvds= f HkT·vds.rc rc

The multiplication by Eh being continuous from (HI(Q»N into itself. this shows that themapplOg

VE(~(Q»N_ ~'c °kT'vds

extends as a linear continuous form (OkT, v)a over the space (HI(Q »N.N

Let 'YI' ... , 'YN be a basis of IRNsuch that T = 2: Ti'Yj with T; ~ O. I ~ i~N, on fen fiti-I

(such a basis exists by hypothesis). Denote by e1' ... , eN the canonical basis of RN and byA E Isom (lR·...) the linear mapping defined by

A'Yi = ej. 1 ~ i~N.

SetN

Sk = 2: (h Tjej E (L\(f c>t·i=\

For v E (H1(Q ))N. one has

Sk . v = 2: Bk Tie; . v = A(BkT) . v = Bk T . A·v,j=1

(3.33)

(3.34)

where A· is the adjoint of A. Obviously, the mapping v - A·v is an isomorphism of (HI(Q »)'"and A·(~(Q))N = (211(Q»)N. From (3.34). it then follows that the mapping

VE(211(Q)t-f. Sk'Vds(=f BkT'A*VdS)(c rc

t A condition obviously fulfilled whenever fen fk = 0.

]346 P. J. RABIER and J. T. ODEN

extends as a linear continuous form (Sk' v)Q over the space CHI(Q»N. In addition

(Sk. v)Q = (Ok T. A *v)Q (3.35)

for every v E (HI(Q »N since equality holds for \' E (~(Q)}"'. On the othcr hand, the com-ponents fhTi. 1 ~ i ~ N. of Sk in the canonical basis of IRs verify fh Ti ~ 0 on fk n f c sinceOk ~ O. As supp Ok C fl,:. one has fhTi ~ 0 on all of f c and lemma 3.4 ensures for everyv E (HI(Q »N that SI,:' vEL I(fc) with

(Sk . v)Q :::: J Sl,:' v ds,rc

Using this result in conjunction with (3.34) and (3.35) in which (A*)-I v replaces v and sincethe mapping v- (A*rl v is continuous from (H1(Q»)'" into itself, we deduce for everyv E (HI(Q»N that 0kT· v E L1(fc) with

(OkT. \')Q:::: f (JI,:T· His, (3.36)r('

Since 2: f)k :::: 1 on f. onc has T· v ::::2: 0kT· \' ELI (f d and the rclationk k

(3.37)

obvious for \. E (QD(Q»;\', rcmains valid for \. E (r/1(Q»N by dcnscness and continuity. Thecombination of (3.36) and (3.37) yiclds

(T.v)::::f T·vus,I'c

for an arbitrary \' E (H1(Q »N and the proof is complete. •

We are tinally in position to prove the following lemma.

LEMMA 3.6. Let T?; 0 be an clement of L I(fd and suppose that the mapping

v E (~(Q»N - f Tv" cis = f Tn' v dsr(' rc

extends as a linear continuous form (Tn. v)Q over the space (HI(Q»N. Then, for everyv E (J-11(Q»"'. one has Tv" E LI(fc) and

(Tn, \')Q ::::f Tv" cis.r('

Proof The boundary f of Q being Lipschitz continuous. there is a covering (fk) of f byopen subsets such that rk is the graph of a Lipschitz-continuous function. This easily impliesthat there is E> 0 such that the angle between two outer norma' vectors n(x) and n(y), x andyin rk, is less than or equal to 1800

- E. or. equivalently. that all the vectors n(x), x E fk• arecontained in a cone with angle less than 1800

• Clearly. this leads to the conclusion that there

,. Signorini-like contact problems through interface models--I

nl II

Fig. I.

1347

is a system of coordinates in which IIi ~ 0 on fk (sec Fig. I whcn N = 2: all the normal vcctorshave nonnegative components in the basis 'YI' 'Y2)'

As TE LJ(fc) one has Tn E (LI(fc))N and. in the appropriate system of coordinatesexhibited above. Til; ~ 0, I ~ i ~ N on fen fk sincc T ~ O. Our assertion is then a simpleapplication of lemma 3.5 with T = Tn. •

THEOREM 3.1. An elemcnt II E (}1I(Q»N is a solution to (3.20)-(3.23) if and only if(P(II")I',, E L1(fC> for every \' E (}1I(Q )).\' and

(I(II,\')+j </J(II")II,,ds=(f,v)u foreveryvE(JfJ(Q))'\'. (3.38)I'c

Proof. Suppose that (P(lln)vn E L l(fd for evcry \' E (HI(Q »)"'. Taking \' as the constantfunction equal to the ith vector of the canonical basis of IR''''.we find

</J( II" )11 i ELI (rd. 1 ~ i~N.

Thus.

</J(Il,,) sup IlliIEL1(fC>.l~i~N

As sup IlIi(X)1 ~ I/VN for almost all x Ere (d. 3.1») wc deducel~i~N

o ~ </J(Il,,) ~ \IN </J(Il,,) ~u'p .lllilI=/~,"

and hence (P(Il,,) E L I(rc). If. in addition. (3.38) holds. it is obvious from lemma 3.2 that u isa solution to problem (3.20)-(3,23),

Conversely. ifu is a solution to problem (3.20)-(3.23) . lemma 3.2 ensures that (P(lln) E L I(rdand the mapping

\' E (~(Q»)N - f </J(IlIl)v" d\'I'c

1348 P. 1. RAmER and J, T. aDEN

extends as the linear continuous form over the space (HI(Q )),\'

v E (H1(Q»N - (f, v)Q - a(u, v).

The conclusion folIows from lemma 3.6 with T = cP(lln)' •

With theorem 3.1 as a starting point. we shalI now be able to prove the following theorem.

THEOREM 3.2. Let u E (HI(Q »N be a solution to problem (3.20)-(3.23). Then. u is a minimizerof the functional

(3.39)

(3.41 )

where j denotes the functional (3.16).

Proof Let u E (HI(Q »N be a solution to problem (3.20)-(3.23). We first show thatleu) < +00. or. equivalently, that j(u,,) < +0::. From (3.9). it folIows that the mapping«Jl(x, t) = fb cp(x. r) dr verifies

to ~ «Jl(x. t) ~ 2 ¢(x, t)

for t > 0 (a relation that was already lIsed in the proof of theorem 2.2). This inequality extendsto t E IR since «Jland ¢ vanish for nonpositive values of t. In particular. taking t = un(x), weget

- I -o ~ «Jl(un) ~ 2 ¢(lln)lln' (3.40)

As ci>(Un)lln E LI(fd from theorem 3.1, we find <b(ll,,) E £.I(rd. namely. j(Il,,) < +00 (cL(3.16)).

Now. Jet v E (HI(Q »N be given. We must prove that lev) ~ leu). Of course, this is satisfiedif j(vn) = +00. Assume then j(vn) < +00. i.e. (vn) E L I(f C). One has

1lev) -leu) = -a(v - u. v - u) +a(u, v - u) +j(v,,) - j(lln) - (f, v - u){2.

2

From our assumptions. the mapping <1>(x.. ) is convex for almost alI x E f eSo that the inequality

«Jl(x, t) ~ <1>(x.1) + ¢(x. r)(t - r)

holds for every pair (t. r) E IR x IR and almost all x Ere. Choosing t = lIn(x) and r = un(x)yields

(vn) ~ (ll,,) + cP(lIn)(V" - II,,).

As the three terms (vlI). (u,,) and ci>(II")(V,, - till) are in vcr d (the latter from theorem3.1 with v - u replacing v). integrating both sides of the above inequality provides

J(v,,) ~J(u,,) + f cP(IIII)(V" - II,,) ds.rc

,. Signorini-Iike conlaCI problems through interface models-I

Applying (3.38) with v - u replacing v, we obtain

j(v,,) ~j(u,,) + (f, v - u)Q - a(u. v - u).

Substituting into (3.41), we deduce

Ilev) - leu) ~ 2'a(v - u, v - u) ~ O.

and the proof is complete. •

1349

Remark 3.1. From the proof of theorem 3.2, it also follows if u and v are two solutions ofproblem (3.20)-(3.23), and hence two minimizers of the functional (3.39), thata(v - u, v - u) = O. which characterizes the difference v - u as an infinitesimal rigid motion(cf. Section 4).

The converse of theorem 3.2 is essentially based on the results of Section 2.

THEOREM3.3. Let \I E (H1(Q »N be a minimizer of the functiooalJ (3.39). Then. u is a solutionto problem (3.20}-(3.23).

Proof. Since the functional j (3.16) takes tlnite values for; = v" and some v E (HI(Q Jr'(for instance, if v E (~(Q»N, because Vn E L"(rd so that $(v,,) E L l(rd. as it foIlows fromtheorem 2.1 (ii) with ~ = 0 and 1/ = vn) one has (un) E L I(r c> when u is a minimizer ofthe functional (3.39). From theorem 2.1 (ii) in which'; = Un and 1/ = 0, we deduce thatCp(un) E L l(rd. Next. for any given v E (qn(Q»N, the normal component "" belongs to LX(rc).Again, from theorem 2.1 (ii) in which'; = II" and '1 = tv", the function

IE R- j(u" + 1IJ,,)

is real-valued and differentiable at the origin with

d f -d j(lIn + Iv,,)lt=1I = CP(un)v" ds.I l' e

Hence, the function

IER-J(u+tv)

is real-valued and differentiable at the origin with

d f -di(u + tv)lt=o = a(u. v) + cp(un)/Jn ds - (f. v).I l'e

Since 1=0 is a minimum for the function (3.42). we obtain

a(u. v) + f 4>(ut,)vn ds - (f. v)o = 0rc

for every v E (~(Q»N and the result follows from lemma 3.2 •

(3.42)

1350 P. J. RABIER and J. T, ODEN

The next section is devoted to proving the existence of minimizers of the functional (3.39)under suitable compatibility condition~ between the applied forces and the geometry of rcand to the study of uniqueness or nonuniqueness of solutions to problem (3.20)-(3.23).

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problems in elasticity with nonlinear friction laws, Inf. J. engng Sci. 24. 1755-1768 (1986).12. NECAS J.. Les Mer/lOdes Directes en Theorie des Eqrcari(J/Is Ellipriques. Masson. Paris (1967).]3. NECAS J.. JARUSEKJ. & HASl.INGER J., On the solution of the variational inequality to the 5ignorini problem

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