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Extremality and compactness results

for elliptic hemivariational inequalities

Siegfried Carl

Martin-Luther-Universitat Halle-Wittenberg

Fachbereich Mathematik und Informatik

Institut fur Analysis, 06099 Halle, Germany

Introduction

Let fi C R^ be a bounded domain with Lipschitz boundary 80,. In this note we consider

an elliptic inclusion in the form

Au + d

j(u) 3 f(-,u) + b, in Q, it = 0, on 8Q, (1.1)

where A : X

—>

X* is a linear, strongly monotone elliptic operator from the Sobolev

space X :— HQ(Q) into its dual X*, /:f2xR—»Risa Caratheodory function,

and b e X*. The multivalued term in (1.1) is given by Clarke's generalized gradient

j : R —f 2

\ 0 of some locally Lipschitz function j : R

—>

R, which is defined by

9ci(s):={CeR|i°(s;r)>Cr, Vr e R}, (1.2)

where j°(s;r) denotes the generalized directional derivative of j at s in the direction r

given by

•or

j{y + tr)-j{y)

J°{s;r) = hmsup ,

!/->s,

tj.0 <•

cf., e.g., [3, Chap. 2]. By using (1.2) and assuming certain growth conditions on

j and / the corresponding weak formulation of problem (1.1) leads to the following

hemivariational inequality (HVI for short)

ue X : (Au, tp-u) + / f(u\ (p-u)dx> (F(u) + b,ip-u), V y? e X, (1.3)

where (•, •) denotes the duality pairing between X* and X, and F denotes the Nemytskij

operator associated with /.

* current address: Florida Institute of Technology, Department of Mathematical Sciences, 150 West

University Boulevard, Melbourne, Fl 32901-6988, U. S. A.

Hemivariational inequalities have attracted increasing attention over the last years

mainly due to its many applications in mechanics and engineering, cf. e.g. [5, 6].

This type of variational inequalities arise, e.g., in mechanical problems when noncon-

vex, nonsmooth energy functionals (so-called superpotentials) occur, which result from

nonmonotone, multivalued constitutive laws, such as for example certain contact and

friction problems, cf. e.g. [4, 5, 6]. In many applications the function j is a d.c.-function,

which means that j can be represented as the difference of convex functions.

Our main goal is to provide sufficient conditions on the d.c.-function j and the non-

linearity / that ensure compactness and extremality properties of the solution set of the

inclusion (1.1), or equivalently of the HVI (1.3).

Notations and hypotheses

In what follows let us assume for simplicity that the operator A is given by the Laplacian,

i.e., Au = — Au. Denote Y := L

(fl) and introduce the natural partial ordering < in Y

by u < w if and only if w

—

u belongs to the cone Y

of all nonnegative elements of Y.

This induces a corresponding partial ordering also in X C Y. We impose the following

hypotheses on the function j and /:

(HI) The function j : R

—>

R is locally Lipschitz and of d.c. type, i.e., there are convex

functions jk

•

—>

R, k = 1,2, such that j(s) = ji(s)

—

J2{s) for all s e R.

(H2) Let djk : R

—>

\ 0 be the usual subdifferentials of the convex functions jk

•

Then one of the subdifferentials djk(s) is assumed to be a singleton for each

s € R, and the following growth condition is imposed:

V £ djk(s) :

\r)\

+ v

\s\ for all s 6 R, k = 1,2,

where

and Vk are some positive constants.

(H3) /:i7xR—>Risa Caratheodory function satisfying for some p e Y

and some

positive constant fj,

\f(x,s)\

<p(x) + n\s\ for all (x,s) € Q x R.

By means of hypotheses (HI), (H2) we derive an equivalent formulation of the HVI

(1.3) which will be useful in our study. To this end we derive the following auxiliary

results.

Corollary 2.1. Let hypotheses (HI) and (H2) be satisfied. Then either j or —j is

regular in the sense of Clarke and the following relation holds:

j(s) = dj

{s)-dj

{s) forallseR. (2.1)

Proof.

The convexity of jk • R

—>

R implies that jk are locally Lipschitz, k = 1,2,

and according to (H2) one of the functions jk must be strictly differentiable. Strictly

differentiable functions, and convex functions are regular in the sense of Clarke, cf. [3,

JO.

p.40].

Since a finite linear combination with nonnegative scalars of regular functions

is regular, it follows in case that j\ is is strictly differentiable that —j = (—ji) + 72

is regular, and in case that ji is strictly differentiable that j = j\ +

(—J2)

is regular.

iProm the generalized subdifferential calculus and in view of the convexity of the jk we

get d

j(s) = dh{s) -

(s),

i.e., (2.1). D

Corollary 2.2. Let hypotheses (HI) and (H2) be satisfied. Then the functional J :

Y -¥ R defined by

J(u) = / j{u(x)) dx, u

Y", (2.2)

is locally Lipschitz and regular in the sense of Clarke, and one has

j°(u;w)dx = J°{u;w), (2.3)

where J°(u;w) denotes the generalized directional derivative of J at u in the direction

w, cf. [3, Chap. 2].

Proof.

By (H2) and (2.1) the generalized gradient d

j satisfies the growth condition

6 d

j(s) :

\r)\

< c(l + \s\) for all s G K,

which together with the fact that either j or — j is regular yields the assertion, cf. [3,

Sect. 2.7]. •

If Jk : Y

—>•

R denote the integral functionals associated with the convex functions

jk, then J(u) = Ji(u)

—

J2(u) for any u EY, and (2.1) holds accordingly, i.e.,

J{u) = dJi{u)-dJ

{u) foranyueF. (2.4)

Due to (2.3) the HVI (1.3) may equivalently be written in the form

(-Au,ip-u) + J°(u;<p-u) > (F(u) + b,<p-u) V> e X, (2.5)

or in its corresponding multivalued form

ueX: -Au + d

J(u) B F(u)+ b in X*. (2.6)

Finally, the regularity of j or — j implies due to [3, Theorem 2.7.5] the relation

J(u) = [ d

j(u{x)) dx, uEY, (2.7)

and thus to solve (2.6) corresponds with finding a couple (u, w) G X x Y such that

w(x) € d

j(u(x)) for a.e. x G 0 and

-Au + w = F{u) +

in X*. (2.8)

Taking (2.1), (2.4) and (2.7) into account, formulation (2.8) of a solution of the HVI

(1.3) leads to the following equivalent definition:

Definition 2.1. The function u G X is a solution of the HVI (1.3) if there are wi, w

Y such that

(i) u>fc(a;) 6 djk{u(x)), for a.e. a: e fi, fc = 1,2,

(ii)

(VuV(p

+ wi

ip)

dx = J

+ F(u))ipdx +

(b,

ip),

Vip G X.

Remark 2.1. Since by (H2) one of the subdifferentials djk is a singleton, to each

solution u the corresponding functions wj. e Y associated with u by (i) and (ii) of

Definition 2.1 are uniquely defined. Furthermore, by (H2) and Corollary 2.1 any w G

j(s) admits a unique decomposition w = w\

—

with

u>k

G djk(s).

Remark 2.2. Since the convex functions jk : R

—>

R, k = 1, 2, in the representation of

the d.c.-function j are locally Lipschitzian and one of subdifferentials djk is assumed to

be a singleton, by [3, Corollary, p.33] it follows that one of these functions is continuously

differentiable, i.e., we have in fact either dji = j[ or dj

= j

- We'll keep the notation

djk even if jk happens to be continuously differentiable, and treat both cases in a unified

way.

Definition 2.2. The function u* is called greatest solution of the HVI (1.3) if for any

solution u of (1.3) we have u < u*, and u„ is least solution if

u„

< u is satisfied. The

least and greatest solutions are called the extremal solutions of the HVI (1.3).

A priori bounds

Let hypotheses (H1)-(H3) be satisfied throughout this section. We are going to derive

a priori bounds for solutions of the HVI (1.3). To this end let us consider the following

elliptic problems:

ueX: -Au + dJ!{u) 3 p + c

+ (fi + v

)\u\ + b in X*, (3.1)

ueX: -Au + dji{u) B -p-c

- (fi +

V2)\u\

+ b in X*, (3.2)

where p

and c

, v

, /i are as in hypotheses (H2) and (H3).

Lemma 3.1. Let Ai > 0 denote the principal eigenvalue of the Laplacian A, and

assume

fj,

+ v

< Ai. Then problems (3.1) and (3.2) have uniquely defined solutions u

and u, respectively, which satisfy u<u and which are a priori bounds for any solution

of the HVI

(1.3).

Proof.

Let us consider (3.1), which means that we are looking for a couple (u,ioi) e

X x Y such that w

(x) G dj-i_(u(x)) a.e. in

and

(VUV(/J-

{n + vi)\u\ip + wiip\ dx = I (p + c

)ipdx+

{b,

ip),

VyeX (3.3)

Define an operator A by

(Au,tp):= / \VuVip - (fi + v

)\u\(p) dx,

then one readily verifies that A

—¥

X* is continuous and bounded. Using Rayleigh's

formula for the principal eigenvalue and taking into account that ||Vu||y = ||u||^ we

obtain the following estimate

{An -Av,u-v)> ||V(u - v)f

- (/, + v

)\\u - v\\

> (1 - ^±^)||V(u -

v)\\

>(1-£±2)||

-„|&,

which shows that in view of n + v

< \i the operator A

—¥

X* is strongly monotone.

Let J\ be the integral functional associated with the convex function j by

•M") = / ji(«(z))d:r,

then by hypothesis (H2) J\ : X C Y

—¥

R is well defined, convex and continuous, and we

have w\ G dJi(u) if and only if wi(x) e dji(u(x)) for a.e. a; € f2, e.g., cf. [2, Theorem

2.3].

Hence, (3.3) may equivalently be written as

u€X: Au + dJ^u) 9 6 in X*, (3.4)

where b 6 X* is given by (6,

<p)

= /

(p +

C2)

tpdx + {b,ip). Since .A is strongly monotone,

it is also coercive, and thus we may apply [7, Theorem

54.

A] which ensures the existence

of a solution u G X of (3.4). The uniqueness of the solution of (3.4) readily follows from

the monotonicity of dJ\ and strong monotonicity of A : X

—¥

X*.

By similar arguments one proves the existence of a unique solution of problem (3.2).

Let u and u be the unique solutions of (3.1) and (3.2), respectively, i.e., we have

(u,«)i),

{u,uii) £ X x Y with wi{x) e dji(u(x)) and w.i{x) 6 dji(u(x)) such that

-Au + wx = p + c

{/J.

V2)\u\

+ b in X*, (3.5)

-Au + uji =-p-c

-(/i +

)|u|

in X*. (3.6)

Subtracting equation (3.5) from (3.6) and using the nonnegative test function ip =

—

u)+ = max(u

—

«, 0) G X

we obtain

/ V(M - u)V(u - u)+ dx+ (i! - w!)(u - u)

dx < 0. (3.7)

Jn in

The second integral on the left-hand side of (3.7) is nonnegative by the monotonicity of

dji, and thus from (3.7) we get

/ \V{u-u)

dx = I V(u-u)V{u-u)+dx<0,

Jn Jn

which shows ||(u

—

||x = 0, i.e., (u

—

= 0, and thus u < u.

Next we are going to verify that any solution u of the HVI (3.1) belongs to the order

interval [u,u], which shows that the unique solutions of (3.1) and (3.2) are in fact a

priori bounds. Let us prove that u < u holds. By Definition 2.1 the function u is a

solution of the HVI (1.3) if there are w^ e Y satisfying w

(x) 6 djk{u(x)) for a.e.

x € Q, k = 1, 2, such that

-Aw +

= w

+ F(u) +

in X* (3.8)

holds.

Due to the growth conditions of (H2) and (H3) we get

««2 < C2 + i>2|«|, F{u)<p + n\u\,

and thus by means of (3.8) any solution u of the HVI (1.3) satisfies (in the weak sense)

the following inequality

-Au + w

< p + c

(fJ.

+ v

)\u\ + b in X*. (3.9)

Subtracting (3.5) from (3.9) and using the special nonnegative test function (u

—

X C\Y

as well as the characterization of the principal eigenvalue, and taking into

account the monotonicity of dji we obtain

Ai [ \{u-u)+\

dx< [ \V(u-u)+\

Jn Jn

< I V(u

—

u)V(u

—

dx + / (wi

—

wi)(u

—

Jn Jn

</(/* + "2)(M - |«|)(u - u)

<(v + v

) [ \(u~u)

dx, (3.10)

which due to /i + v

< Ai implies (u

—

0, and thus u < u. in a similar way one

proves that u < u holds, which completes the proof of the lemma. •

Remark 3.1. Applying the comparison technique used in the proof of Lemma 3.1 one

easily shows that any upper solution v of problem (3.1) satisfies v > u and any lower

solution

of (3.2) satisfies v < u, and hence v and

are also upper and lower a priori

bounds of the HVI (1.3). In general, it is easier to find upper and lower solutions for

(3.1) and (3.2), respectively, rather than to solve the corresponding problems exactly,

which is useful in applications.

Main results

Theorem 4.1. Let hypotheses (H1)-(H3)

fulfilled, and assume fi+v

< Xi, where Ai

is the principal eigenvalue of the Laplacian A. Then the HVI (1.3) possesses extremal

solutions, and the solution set is compact in X.

Proof,

(a) Existence of extremal solutions

We are going to prove the existence of the greatest solution. Let u and u be the a priori

bounds given by Lemma 3.1, and let h

: R

—>•

R be the increasing function which is

related with the subdifferential dj

•

R -4 2

\ 0 by

{s) = [h

{s),h

(s)},

where h

(s) and h

(s) denote the left-sided and right-sided limits of h

in s, respectively.

Consider the elliptic problem

ueX: -Au + dj^u) 3h

(u) + f(-,u) + b in X*. (4.1)

Since by (H2) we have |/J2(s)| < c

+ v

\s\, one readily observes that u and u are lower

and upper solutions of (4.1), respectively. Introduce the multifunction

x R x R

—»

\ 0 defined by

P(x,r,s) :=9ji(r) -

f(x,s),

then (4.1) may be rewritten in the form

-Au + /3(,u,u) 3 h

(u) + b in X*, (4.2)

where /3 may be considered as a state-dependent maximal monotone graph. Since u

and u are lower and upper solutions, respectively, of (4.2), we may apply the theory

developed in [1, Chap. 6.1], and, in particular [1, Theorem

6.1.2]

ensures the existence

of a greatest solution of (4.2) within the interval [u,u]. Let

:= u and define an

iteration process as follows: u

n+1

G X is the greatest solution within the interval

[u,

]

of the following problem

u€X: -Au + /3(-,u,u)3h

) + b in X*. (4.3)

As seen above this process is well defined for n = 0, which yields ui as the greatest

solution of (4.2) in

[u,

]. Let n = 1, and consider the following problem:

ueX: -Au + /3(-,u,u) 3 h

(u{) + b in X*. (4.4)

Since h

is increasing and u\ <

UQ,

we get /i2(wo) > h

{u\), and thus ui is an up-

per solution of (4.4). One readily verifies that u is a lower solution for (4.4). Thus

again by applying [1, Theorem

6.1.2]

there exists the greatest solution of (4.4) within

[u,

ui], denoted by u

. By induction the iteration process is well defined and yields a

decreasing sequence (u

) such that

[u,

]. By definition these iterates satisfy:

n+1

lin+1

) £ X x Y with w

lin+1

(x) G 9ji(u

i(a:)) such that

/ Vu

iVcpdx+ / wi

in+

iipdx = / (h

) +f(-,u

i))ipdx+{b,ip),

Mip

e X.

Jn Jn in

(4.5)

By means of (4.5) the boundedness of (u

) in Y along with the growth conditions (H2),

(H3) imply its boundedness in X, and thus due to the monotonicity of the sequence we

obtain the following convergence of the entire sequence: u

—*•

u* in X and u

—>

u* in

Y. Since h

is an increasing and right-sided continuous function, we get in view of the

monotonicity of the sequence (u

) and by applying Lebesgue's dominated convergence

theorem

/ h

)<pdx^t / h

{u*)ipdx, as n

—¥

00. (4.6)

Jn Jn

By (4.6), and the weak convergence u

—*•

u* in X, and the continuity and boundedness

of the Nemytskij operator F : Y

—>

Y due to (H3) we get from (4.5) for all

6 X:

lim / w

ln+1

<pdx= - / Vu*V(fidx+ / (h

{u*) + f(-,u*)) (pdx + (b,ip). (4.7)

^°°Jn ' Ja Jn

Prom (H2) and the boundedness of (u

) in Y it follows that (wi,

) is bounded in Y,

and thus there is a subsequence (w-ij) which is weakly convergent in Y. Since X is

dense in Y from (4.7) we deduce that each weakly convergent subsequence of (wi

)

must have the same weak limit, and thus the entire sequence (»!,„) must be weakly

convergent, i.e., uii

—*•

to*. Prom wi

(a;) 6 dji(u

(x)) and the convergence properties

of (wi,

) and (u„) we easily get W[(x) e dji(u*(x)). Setting w

•= h

{u*) e dj

{u*)

we see that the limit u* satisfies

[ (Vu*V<p + wl<p)dx= [ {wt + f(-,u*))

<pdx

(b,<p),

V^el, (4.8)

Jn Jn

where wl 6 djk{u*), k = 1, 2, and thus u* is a solution of the HVI (1.3) according to

Definition 2.1.

Next we are going to show that u* is in fact the greatest solution. To this end let u

be any solution of the HVI (1.3), then u e

[u,

u],

i.e., u < u

= u, and satisfies

-Au + w

= w

+ F{u)+b in X*, (4.9)

where uik e djk(u), k = 1,2. Since w

< h

(u) < h

(uo), from (4.9) it follows that

it is a lower solution of (4.2) defining the iterate u

and trivially u

= u is an upper

solution. Thus there exist solutions of (4.2) within

[u,

]. However, by definition u

is the greatest solution of (4.2) less than

UQ,

which shows u < u\. By induction one

can show that u < u

for all n and thus u < u*, i.e., u* is greatest solution of the

original HVI (1.3). The proof for the existence of a least solution u* can be done in a

similar way replacing /i

by h

in (4.3) and starting with «o := M, where w„+i is the

least solution within

,u].

(b) Compactness of the solution set S

By (a) the solution set S of (1.3) is nonempty. Let (v

) C

be any sequence of solutions,

i.e., (»

,iBi

,iB2,n) £ X x Y x Y such that

-Av

+ w

= w

,„ + F{v

) + b in X*, (4.10)

where uifc

e djk{v„). Let c > 0 be a generic constant. Since ||i>

||y < c for all

it follows that ||wfc,

||y <

f°

r an n an

d k = 1,2, which by using (4.10) implies

||«n||x < c. Thus there exist subsequences (u,),

(u>k,j),

k = 1,2, satisfying Vj

—*•

v in

X, and due to the compact embedding X C Y, Vj

—>

v in y, and w^j -

»t in Y with

Wk(

) 6 9jfc(v(a:)). Replacing in (4.10) n by j we may pass to the limit as j —> oo,

which shows that the limit v is a solution of the HVI (1.3), i.e., v G «S. Finally, the

strong convergence of Vj -> JJ inX follows from (4.10) replacing n by j and taking as

special test function

= Vj

—

v which yields

hi ~ v\\x = I l

(^ ~v)\

dx = - [ VvV(vj - v) dx

+ / (w2

j-u>i,j + f(-,Vj)){vj-v)dx+(b,Vj-v). (4.11)

The weak convergence of (VJ) in X and its strong convergence in Y along with the

boundedness of

(wf.,j),

k = 1, 2, in Y imply that the right-hand side of (4.11) tends to

zero,

which yields Vj

—>

v (strongly) in X, and thus the compactness of S. D

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