Hemivariational inequalities with functionals which are not locally Lipschitz

Voinnlmmr Amdwrr,

Thev>n,. Metho&

Pergamon

& A,,,,lrcolrons. Vol. 25, No. 12, pp. 1307-1320, 1995 Copyright 0 1995 Elsewer Science Ltd Prmted in Great Britain. All rights reserved 0362-546X/95 $9.50 + .OO

HEMIVARIATIONAL INEQUALITIES WITH FUNCTIONALS WHICH ARE NOT LOCALLY LIPSCHITZ ZDZISLAW Umversity

of Warsaw,

Institute

of Applied

NANIEWICZ

Mathematics Poland

and Mechanics,

Banacha

2,02-097

Warsaw,

( Received II) June 1993; received for publication 13 September 1994) Key words and phrases. Hemivariational linearity.

inequality.

directional

growth

condition,

discontinuous

non-

1. INTRODUCTION

In this paper we shall deal with the following nal inequality

(Au-f,u--)I/+

problem

(P): find u E V satisfying

;“(u.u-u)d-n>O

hemivariatio-

VUEV.

I’ is a reflexive Banach space compactly imbedded into LP(R; RN>, 1

denotes Clarke’s directional differential with respect to 5. Our study will be carried out under the following main hypotheses which will be referred to as the directional growth conditions. There exists a constant 1 IS

A: V-t

VIE R” and a.e. x E 0,

j”(x;&,-[)sk(x)l[l

and j”(x;5.77-5)
+l(l’)

V(, qeRN

with lq)~r,

r > 0, and a.e. x E a.

In spite of the fact that the foregoing directional conclusion that the corresponding functional

growth

conditions

do not

allow

the

is locally Lipschitz and even finite on the whole space V, they are sufficient to establish the existence of solution to (PI without any additional requirements concerning j. The notion of being a solution of (PI needs only to be modified. We introduce the following definition. An

1308

Z. NANIEWICZ

element u E I/ is said to be a solution of (PI if there exists x E L’(fi; R”‘)n I/* such that (Au-f,u),,+

x.udIl=O

VUEI/,

/n x(x)

for a.e. x E R,

E 2j(x,u(x))

where ~?j(x,. > stands for the generalized gradient of j(x,.>. To make the foregoing estimates more familiar let us mention that if the function RN 3 5 -+ j(x, 5) satisfies the relaxed monotonicity condition of the form (5* - nY)*(5-

77) 2 -cp(x,r)l5-

Vt, FERN with InI
71

r > 0, and a.e. x E Cl,

where 5 * E Jj(x, 5 1 and n* E c;li(x, n), for some cp: III x R+ + R’, then the corresponding directional growth condition holds with s = 1. To establish the existence of solutions to (P) the Galerkin approximation method will be employed. We use the Kakutani fixed point theorem for the treatment of finite dimensional problems. In order to pass to the function spaces we shall apply the Dunford-Pettis precompactness criterion in the L’-space combined with the finite intersection property. In comparison with the approach given in [l], where the directional growth condition was assumed as j”(x;5.*,-5>1cy(x,r)(l

+I511

V~,nERNwithlnl
and where the regularization of j in the form of the convolution j * we, wE being a mollifier, was used, in this paper no smoothness technic is applied. The main advantage of this fact is that we may relax the foregoing directional growth condition for r > 0 and replace it by jO(x; 5,~-

5) 5 a(x,r>(l

+ 151”)

V(, vERN

with lqlsr,

r > 0 and a.e. x E R,

with parameter s E [l, p). The study of hemivariational inequalities started with the works of Panagiotopoulos [2-51, who by applying the generalized gradient of Clarke [6] and Rockafellar [7], introduced such variational expressions to discuss and solve mechanical problems with nonconvex and nonsmooth energy functionals (problems with nonmonotone laws and boundary conditions, nonmonotone stick-slip and sticktion effects, sawtooth laws in composite structures, nonmonotone semipermeability problems, delamination of multilayered plates, partly debonding of adhesively bonded cracks and of adhesively connected elastic bodies, nonmonotone friction contact prolems, etc.). Both the mathematical and mechanical aspects of the theory of hemivariational inequalities involving a function j expressed as a primitive, i.e. j(t)= /0

c P(r)dr

can be found in [4,5] and [g-12]. In [13] coercive and semicoercive hemivariational inequalities have been derived for which necessary and sufficient conditions for the existence of solutions has been formulated. The general case in which J is assumed to be locally Lipschitz

Hemwariational

inequalities

with function&

1309

which are not locally Lipschitz

on V and A + dJ is coercive was treated by means of the pseudo-monotone mapping theory in [14-171. The hemivariational inequality approach to constrained problems with star-shaped admissible sets can be found in [18]. For the related topics the reader is referred to 119-251. Let I/ be a reflexive Banach space and let 0 c R" be an open bounded subset of R",n 2 1, with sufficiently smooth boundary. Throughout the paper it will be supposed that the imbedding VcLp(Q R”), N 2 1, is compact for some 1

Let j: 0 x R,’ --) R be a function defined for almost all x E Q and all 5 E RN, which is assumed to satisfy the following conditions (i) for all 5 E RN the function sZ3x-j(x,[) (as a function of the variable x> is measurable on R, (ii) for almost all x E fl the function RN35-jj(x,[)

(as a function of the variable 5) is locally Lipschitz on R”. The properties (i) and (ii) ensure that j is a Carathiodory function. Since j is assumed to be locally Lipschitz with respect to [, for almost all x E 0 the Clarke’s directional differential can be defined according to the formula [6] j”(x; {. 77)= limsup h + 0 A \i 0

It permits the formulation

j(x;[+h+Aq)-j(x,5+h) A

6, +RN.

of the generalized gradient of Clarke as

dj(x;e)={vERN:

j0(_r;~,y)2~.y

be

RN}t

for almost all x E f1 and for all 5 E R”. It is important in our study to ensure the integrability of j(., u(. 1) and j”(.; u(. >,u( . )) for any U, u E V n L”(R; RN). Therefore, it will be also supposed that there exists a function p: 0 x R _ 4 R fulfilling the properties (1”) p(..r)EL’(~1)foreachr20:

(1)

(2”) If I’ 5 r” then for almost all x E 52, /l
I p(x,

r”);

and such that l;(+.5)-j(x,77)ljp(x,r)l5Vhroughout the respectively.

paper

the symbols

ql

Vt, 71EHO,r),r2 0,

1.1and u .” denote the norm and the inner product

in the Euclidean

(2) space

RN,

1311

Hemwariational inequalities with functionals which are not locally Lipschitz

Let A be the family of all finite-dimensional subspaces F of I/ fl L”(Q; RN), ordered by inclusion. For any f~ A we formulate the following finite dimensional problem. Problem (P,).

Find Us E F and xF E L’(f2; R” 1 such as to satisfy the following relations (Au,-f,~)~+

VUEF,

/,,.,:dfl=O

(6)1

bl

and for a.e. x E a.

XF(X) E a;cx; up(x))

(6)~

To begin with let us make some observations. If F E A and uF E F then from (1) and (2) it results P(x,IuPI + l)lwl I P(x,IIu,ll~~~n~ + l)llwll~-~nj

ljO(x; u,tx),w(x))lI

for any w E L^(R; R&Y)and for a.e. x E 0. Hence 1 ~j%,.wb+

~/3Ua,llr,I,,+

I~dftllwil~~~o~

(7)

k’w~L3ft;R~).

+ 1) E L’(Q) by Cl),, the integral on the left-hand side of (7) is finite. It Since P(IIu~IIL-~~~, allows us to define the following mapping I,.: F 4 2L”n. R”) by the formula IF(cf)

= /$ELI(G: \

RN):

($.

WdiI 5 [jO(UF.~)dfI

JIl

Vw~L=(fln;R~)

J11

I, I

(8)

for each uF E F. It follows immediately that if (1,E IF(up) then $4~) E aj(x; U,(X)) for a.e. XES1.

LEMMA 3.1. Let LJ~E F for some FE A. Then compact subset of L’(0: R”“‘).

IYuF 1 is a nonempty

convex and weakly

Proof: Since R” 3 5 -j(x, 5 ) is locally Lipschitz, dj(x, u,(x)) is not empty for a.e. x E 0. Setting J/(x) E r?j(x, u,(x)) for a.e. x E R, we obtain a function which due to (7) and (8) is an element of Uu,). The convexity of T(u,) follows immediately from the fact that the mapping L”(R; RN) 3 H’ + j’(u,,w) is convex. To show the weak precompactness of I’(u,) in L’(R; RN> choose arbitrarily 9 E It uF). Then from (7) it follows (c/. u’dl(l I j”(u,,w)dfl /0 / I1

5

P(II~FIIr.‘m + 1) dfW+,=~~~~

/ I1

fo,r any L”(i2; R”). Let us denote by 6 an element from L”(iR; RN) with the properties that

II~IIL=~o,I v@ and

1312

for a measurable

Z.

NANIEWICZ

o c il.This easily implies (9)

Since,

for any E > 0 there exists 6 > 0 such that

l$ldfI 5 fl

Iw

P(IIuFIIL=(I~) + l)dfi 5 E, /w

whenever mes(w) < 6. The Dunford-Pettis criterion for the weak precompactness in L’(Q) holds f26]. Finally, it remains to show that IYu,) is weakly closed. For this purpose let us suppose that a sequence l&J c Ru,) converges weakly to Cc,in L’(0; RN). Then we have tin.wdCl< / i1

j”(u~,r++)dR / Ii

VW E L”(0; RN),

from which by passing to the limit n + 3~we arrive at I/J. wd0 2 /0

j”(u,,w)dR / I1

VW E L”(R; RN).

It means that 4 E I(L.~) and the proof of lemma 3.1 is complete.

n

For FE A, let i, : F - V denote the inclusion mapping of F into V, and i: : V* + F* the dual projection mapping of I’* onto F*. The duality pairing over F* x F we shall denote by (. , . )F. Further, let 7F : L’(R; R”) --) F* be an operator which assigns to any 1(1E L’(% RN> an element TV4 E F* defined by (10) Now consider a mapping T, : F + 2F* given by formula

T,(o,)

= TFrF(uF)

for LJ~E F.

(11)

The following properties of TF can be pointed out. 3.2.TF : F --) 2F’ is a mapping from F into 2”* such that for each vF E F, TF(uF) is a nonempty bounded closed convex subset of F*. Moreover, TF is upper semicontinuous from F into 2F*. LEMMA

Proo$ It is easily seen that 7F defined by (10) is linear and continuous from the weak topology in L’(R; RN> to the (unique) topology on F*. Thus from lemma 3.1 it results that for each uF f F, T,(u,) = TFrF(uF) is a nonempty bounded closed convex subset of F”. It remains to show that TF is upper semicontinuous from F into 2F*. For this purpose let us

Hemivariational inequalities with functionals which are not locally Lipschitz

1313

sequence suppose that a sequence Iu,,} c F converges to uF in F, and that the corresponding (T~I)~} cl;* with 4n E rF(uFn), converges to some I)* E F* in F*. We have to prove that there exists I)‘E rF’,(u,) such that $* = T~I)‘_ First we notice that the convergence of {up,,) in F implies the existence of a finite upper bound

for {IIu~~II~=(~~J, say C, i.e.

IIUf+llL”(~1) 5 C‘ Taking

into account

that

/ iI we obtain

3 ) L)....

forn=l

VW E L”(fl;

;“(v,,,w)dO I I1

‘4” .wdR<

RN),

(12)

the estimate

I

I$nldfI 5 i%

which due to the integrability 6 > 0 with the property that

p(C + l)dfi,

wCR,

Iw

w

of p(C + 1) allows us to say that for any E > 0 there

exists a

provided mes(w) < 6. On the basis of the Dunford-Pettis theorem we assert the weak precompactness of { I,$,} in L’ (Q; R” ). Accordingly, a subsequence of 1I/J,) can be extracted (again denoted by th e same symbol) such that I,!J,,+ I,!J weakly in L1(fl; RN) for some 4 E L’(0; R”). It implies that TV(6;,+ 7F CC, = $* in F*. We claim that $E I’,(u,). Indeed, taking into account the upper semicontinuity of the function

by virtue

of (12) we are led directly $. wdfl< I 11

which is equivalent

to the statement

tf~~L”(ln;R”),

j”(u,,cr)d0 / 11

to $ E I’( ufi ). as desired.

Now we are ready to pass to the existence

that (13)

n

result (P,).

PROPOSITION 3.3. For every F E :I the problem (P,> has at least one solution (u,, xF) E F X L’(ln; RN)? i.e. (6), and (6): hold. Moreover, the family {u,}~,,, is uniformly bounded, i.e. there exists a positive constant M not depending on F E A such that

Proof Define A, = i:. Ai, and fF = iF,f’. Notice that in order to establish the solvability of (P,) it suffices to show that ffi belongs to the range of the multivalued mapping A, + TF. We shall show that. in fact, range (A, + TI; 1 = F*. Indeed, in view of lemma 3.2, for each u E F,

Z. NANIEWICZ

1314

A,(u) + T,(U) is a nonempty bounded closed convex subset of F*. Further, the weak continuity of A from 1/ into V* implies the continuity of A, from F into F*. Therefore, due to lemma 3.2 the mapping A, + TF is upper semicontinuous from F into 2fm. Moreover, on the basis of (4),, (7) and of the coercivity of A we arrive at

2

c(llullv)llullc~-

-kbldR 2 c(lbllv)lbllv/ II

2 c(llullv)llollv- IlkllL~~n,yll&,

ll~llucn,ll&qn,

VVEF,

where l/p + l/q = 1, and y is a positive constant with the property that IIullL~(n)I Yllullv, VU E V(V is compactly imbedded into L P(fl; RN ). This means that A, + TF is also coercive. Now it remains to invoke the Kakutani fixed point theorem in the version for coercive mappings ([27, proposition 10, p. 2701) and thus we may conclude that range (AF + TF) = F*. Accordingly, for f~ I/* there exist uF E F and xF E L’(O: RN> such that fF EAT + T(u,), i.e. (6) holds. To show that the family (~~1~~ , is uniformly bounded we make use of the estimates

2 (Au, ,U,>L._

j”(uF, -u,)dfi / II

~c(llu,ll~)llu~ll~-

In

klu,ldfl

2 c(IIu~lIv)IIu~lIv - llkllr~r~r~,ll~~II~~~r~, 2 c(II~~IlvN~FllY - Il~llL~cn,Yll~Fllv~ which thanks to the well-known properties assertion. n The next lemma corresponds

of the coercive function c = cc. ), establishesthe

to the compactness

property

of the set { xF: F E A) in

L’(Q; R” ). LEMMA 3.4. Let for any FE A

a pair (u,, xF) E F x L’(R; RN) be a solution of (I’,). Then the set { xF: F E ,\I is weakly precompact in L’(iZ: R” 1. ProoJ According to the Dunford-Pettis theorem it suffices to show that for each E > 0 a 8, > 0 can be determined such that for any w c R with mes(fi) < S,,

I ,yFldfl

< E.

FEA.

/w

Fix I > 0 and let 77E R” be such that 1~1sr. Then wehave

xF(x). (7 - u,(x)) 2 j”(x; u,(x), 77- u,(X))

(15)

Hemivariational

from which, by virtue

inequalities

with functmnals

which are not locally Lipschitz

131s

of (4&, it follows that &_(X).

for a.e. x E f1. Denoting

q 2 ,I&)

by x~,(x),

u,(x)

i = 1, 2,.

77(x)= +(sgn

+ dx;

r)(l

+ lu,(xN”)

. . N, the components

of ,Y&)

we set

,yF,(x) ,..., sgn xFhi(x)),

where

sgn y =

1

ify>O

0

ify=O

i -1 It is not difficult

to verify that /q(x)l~

x/,(x). Therefore,

0.

for almost all x E fl and that 7/(x) 2 rlfi

&A-r)/.

we are led to the estimate

*I Integrating

r

if Y <

this inequality

&(X)1

I

/yf-(x). u,(x)

over wcR

+ a(x;r>(l

+ lu,(x)l”).

yields

Thus, from (14) we obtain

(16) Now we show that (17) for some positive

constant

C not depending

on o c 0 and

F E A. Indeed, from (4), one can

easily deduce that XF(X).UF(X)fk(X)(l

+lu,(x)l)20

for a.e. x E a.

1316

Z. NANIEWICZ

Thus we can write that

([

XF

.

uF-

k(l

+

+

lu,cI>l da 2 ( [ xF uF + kc1 + l+I)l da, JCl

Jw

and consequently

+ IlkllLqcn,mes(R)“P However, A as being weakly continuous, thanks to (6), and (14) we conclude that xF / (1

uF dil = -(Au,

From the last two estimates one obtains for r > 0 1x ldfl S EC F

/w

maps bounded

sets into bounded

-f‘, u,->I I IIAu, -fll~~II~~ll~

we easily arrive at (17), as desired.

+ ~Ilu(r)lll,,,,rl,mes(w)‘lp

r

+ IlklI~.4~~~yM.

(18)

sets. Therefore,

(2 = const.

I 6

Now, when applied

(17) to (16)

+ ~Iln(r)llrq,cw,y”MS. r

r

(19)

(20)

NOW, let t > 0. Fix r > 0 such that

Since

a(.,

r)

E

L’7

( 121, we can determine

6, > 0 small enough

dv + ~Ila(r)ll~4,,~ySM”

~I,rr(r)lll.~~nlrnes(w)“” whenever

mes( WI < 6,. Finally.

THEOREM

5 5

it yields

for any w c 0 with mestw) < 8,. Accordingly, L’(0: R”) is established. n Now we are in a position

so that

to formulate

the weak precompactness

our main existence

of { XF: FE

A} in

result.

3.5. Let A be a coercive, weakly continuous operator from V into V*. Suppose that the injection VcLP(R; R” ), N 2 1, is compact for some 1

1317

Hemivar~ational inequalities with function& which are not locally Lipschitz

Proof In order to prove the theorem we have to show that there x E V* (7 L’(fi; RN 1 such that (5) holds. For FE h let W,=

U {(uFs,,yf,)) F’EA

exist u E V and

cVxL’(i2;RN),

F’3F

where a pair (u,,, xF.) satisfies (P,,>. We use the symbol weakcl (W,) to denote the weak closure of W, in I/ x .!,I( (1; R” >. Moreover, let Z=

U {x~} cL’(O;R’V. Ft A

Denoting by weakcl (Z) the weak closure of Z in L’(fZ; R”), from (14) we obtain weakcl cB,(O,

M) x weakcl (Z)

VFER.

Since B,(O, M) is weakly compact in V and, by lemma 3.3, weakcl (Z) is weakly compact in L’(R; RN>, so the family {weakcl : FE A} is contained in the weakly compact set BJO, M) x weakcl (Z) in I/ x L’(IR; R”). Now, on the basis of proposition 3.4 it results that this family has also the finite intersection property. Thus the intersection 0 FE

( W, )

weakcl .Z

is not empty. Let (u, x) be contained in this intersection. The proof will be complete if we show that (u, ,v) satisfies (51, and (5,). Let u E V fl LYR; R”) be arbitrarily given. We choose FE .A such that u f F. There exists a sequence {(UF,,,XF,,1 in W,, (for simplicity of the notations we denote it by (un, x,1> converging weakly to (u, x) in I/x L’(R; R”). It means that Ll, + u

weakly in I/.

(21)

weakly in L’(R; R”).

cm

and x, + x

Moreover, the following equality holds for n = I, 2,. . (Au,-f,v)b

+

( xn.udR=O.

(23)

Now, taking into account (21) and (22) we can pass to the limit as n + m in (23). The weak continuity of A and the arbitrariness of L‘ E G’ n L”(R; R,‘) permits us to conclude that the equality (Au _f’,V)l

+

X.“df2=0 / Ii

(24)

is valid for any L’E V fj A”(Q: Rx ). However, by the density of Y n L”(fi; RN >in V we finally arrive at (5),. The last step of the proof consists in proving (5),. For this purpose let us notice that due to the compact imbedding Vc LP(fl; RN>, from (21)-it results that U” ‘U

strongly in L”(l); RN),

(25)

1318

Z. NANIEWICZ

which without

loss of generality

allows us to assume K, ‘U

with u E Lx(O - w: R’).

Accordingly,

uniformly

weak

for any E> 0 a subset

convergence

of

w C fl with

on 0 - 0

- w; RN) be arbitrarily

Let u E L”(0

x,, . v dll I / SIPw combined with the semicontinuity of

(26)

a.e. in a.

Thus we can apply the Egoroff’s theorem. mes(o) < E can be determined such that U” ‘U

that

(27) given. From the estimate

j”(u,,u)dfi / s1- w x,

to

x

in

(28) L’(0;

RN),

(25)

and

the

upper

in LP( Sr.. R” 1, we obtain j”(u,u)dl(l / II-w

X.vdil< / il&w However,

the last inequality

implies

x(x) where

mes(w)

< e, therefore,

x(x) i.e. (5), holds. This completes

- w; RN).

that

E Jj(x; since

VU ~L=(fi

u(x))

E was chosen

E Jj(x;

u(x))

f0ra.e.

xEfi-w,

arbitrarily fora.e.xEfi,

the proof of the theorem.

n

To close the paper let us make some comments. First of all we pay attention to the qualitative difference between the cases N = 1 and N > 1. As it has been shown in [l], if N = 1 then the directional growth condition (4), implies (4), with s = 1. So, in such a case, (4), is redundant. When N > 1 then the condition (4), is necessary. It is shown in the example below. Example 3.6. Let us consider

where

m is a natural

number.

a locally Lipschitz

Then

function

j: R* + R defined

one can easily check that

j”([.-050

V&-ER2,

as follows

1319

Hemivariational inequalities with functionals which are not locally Lipschitz

i.e. the condition

(4), holds with k = 0. On the other hand, for 77= (0, - 1) and .$* > 1 we have

from which it follows that we cannot

obtain

a better

estimate

than that of

with the power 2m. It is not difficult to observe that the growth condition (4), has been used, in fact, to establish the boundedness of the approximation sequence {u,) in I/. If more informations on A are available then this condition can be weakened. For instance, if A is strongly monotone and p 2 2 then instead of (41, we are allowed to suppose that QE

j”(x;5,-5)~k(x)(l+l~l”) with 1 I (+< 2 and k ELq(fi; RN), possible to consider the estimate j”(x;

where

5, -0


ij=p/(p

- rr).

Moreover,

that s
V(,TER”

j”(.r:~,77-_5)IcY(X,r)(l+151P)

if p>

2 then

arises what can

withqEB(O,r),

r > 0, and a.e. x E a,

with a(-, r) E L”(0). r > 0. In order to get an exsitence requirement is needed. Namely, if we assume that

result

(29)

in such a case an additional

lim Ila(r)ll~.~cil,/r = 0 *‘= and take into account

/ 0J

that (20) can be written

I ,yF.IdiZ I EC r

we can easily deduce

(30)

then as

+ ~,,u(r)ll,.,,~,,rnrs~a,)

that lemma

it is

VJ5E R”,

+ 61~1’

with C E L’(i1) and sufficiently small E > 0. In the growth condition (4), it has been supposed be said about the limit case s =p, i.e.

RN,

r

+ Elia(r)ll~-c r

‘U,yPMP,

3.3 still holds. We then arrive at the following

result.

THEOREM 3.7. Let A be a coercive, weakly continuous operator from V into V*. Suppose that the injection I/c L’Yfl; R”). N 2 1, is compact for some 1

has at least one solution. RI:FERENCES

I. NANIEWICZ 2.. Helnivarlational Analysir (to appear) 2. PANAGIOTOPOULOS Res. Cummuns

inequahtles

wth functions fultilling directional growth condition,

P. D., Nonconvex superpotentials

8, 3.15 340 (1981).

Applic.

in the sense of F. H. Clarke and applications. Mech.

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Z. NANIEWICZ

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