The lack of lower semicontinuity and nonexistence of minimizers

Nonlinear Ano/.vsrs, Theory, Methods & Applicofrons, Vol. 23, No. 1, pp. 143-153, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britam. All rights reserved 0362-546X/94 $7.00+ .OO

Pergamon

THE LACK OF LOWER

SEMICONTINUITY OF MINIMIZERS-f

AND NONEXISTENCE

FABIAN FLORES~ International (Received

School

for Advanced

22 December

Studies,

Via Beirut 4, 34014 Trieste,

1992; received for publication

Key words and phrases: Lower semicontinuity, conjugate function, functions of integral type.

convexity,

subdifferential,

Italy

4 June 1993) relaxed

function,

minimizer,

1. INTRODUCTION

IN THE STUDY of minimization problems in the calculus of variations, the lower semicontinuity (1.s.c.) property, or a weak version of it, plays a prominent role in the direct method. Recently, nonlower semicontinuous functions or nonconvex integrals (for the so-called scalar problems) have been areas of active investigation (see [l-6] and references therein). We are not aware of any general theory in this direction. However, some existence results of minima for nonconvex integrals have been given in particular cases, namely, when the integral can be split into the sum of two integrals, one depending on the derivative of higher order (e.g. gradient, Laplacian), and the other depending on the state function. Thus, sequentially weakly continuous (s.w.c) perturbations of integrals that are not sequentially weakly lower semicontinuous (s.w.l.s.c.), i.e. the nonconvex part, are actually considered. Then, under ad hoc assumptions on the continuous term, one gets the desired result. The purpose of this paper is to contribute to a better understanding of the phenomenon that arises when the 1.s.c. or the convexity property fails. In the next section (theorem 2.4), given a non-1.s.c. function, we show the existence of nonnegative continuous functions: perturbations of the first, such that the perturbed function does not attain its minimum. Moreover, such a perturbation can be chosen as small as one wants under some additional assumptions (theorem 2.4 and remark 2.6). The argument used here is similar to that in [3], where the author establishes a relationship between the nonexistence of minima for nonconvex integrals and the uniqueness for the convexified (relaxed) integral, which is not strictly convex. In Section 3, we deal with the simplest integrals depending either on the gradient or on the Laplacian. For either case, we are able to construct continuous perturbations in an integral form (theorems 3.4 and 3.6). However, we have to point out the difficulty presented here, since we do not allow the boundary datum to vary as in [7, theorem 5.11. 2. THE

ABSTRACT

CASE

In this section X is a metric space. For r > 0, B(u, r) denotes u with radius r, and &u, r) the closed ball.

the open ball in X centred

at

t The results of this work were presented at the Third French-Latinoamerican Congress on Applied Mathematics and First Chile-CEE School on Optimization, held in Santiago from 28 August to 5 September 1992. $ Permanent address: Universidad de Conception, Departamento de Ingenieria Matematica, Casilla 4009, Conception, Chile. 143

F. FLORES

144

PROPOSITION 2.1. Let @: X + [0, fco] be a lower semicontinuous function, and let u. be in X such that cD(u,) < +a. Then there exists a nonnegative continuous function Y defined in X such that Y(U) 5 Q(U), B(uo) = Y(u,) 2 Q(u) for u E X. Proof. Set for every n E N a,, s inf(Q(u): u E B(u,, l/n)]. Clearly a, 2 0 and (a,) is a nondecreasing sequence bounded from above by Q(uJ. The lower semicontinuity of Q implies that a + @(z+,) = sup,, a, = lim, a,. Let us consider for every n E N a continuous function p,, defined in X, satisfying 0 I q,(u) i 1 for u in X, ~1, = 1 in @u,, l/(n + lj) and p,, E 0 in X\B(u,, l/n). Define the function Y as follows

+cm

(an+1- an)vn+l@)+

Y(u) =

al

if u E B(u,,

l

l/2)

n=,

otherwise.

al ~~(4

This function will satisfy our requirements. First of all, notice that Y is well defined: it becomes a finite sum for every 24 in B(u,, +), Y(U) = 0 in X\B(u,, l), Y(u,) = a = @(uo), since p,+i(u,J = 1 for all n, and 0 I Y(U) I a = Q(u,J VU E X. (a) We shall prove Y(U) 5 CD(u) v u E X. If t? $ B(u,, 3), then either Y(c) = 0 or Y(fi) 5 a,, therefore Y(ti) 5 t. On the other hand, it is not difficult to prove it at d with d(li, u,) = $. We only check it at uO. Let E > 0, then by the convergence of (a,) there is fi E N such that 0 < a - a, < E Vn 1 fi. The definition of continuity at u0 is verified by taking 6 5 l/(fi + 1). Indeed, if u is such that d(u, uO) < 6, then p,(u) = 1 for n = 1, . .., fi. Hence, IY(u) - Y(uO)/ =

C (a,+, n

- a,)~,+,(4

+ a1 -

C (a,+, n

C (an+1- a,)(1 - v7,+&4) = C (a,,, -

=

nrn

n

C

5

(a,,,

- a,) + ai

II

a,)(1 - ~,+l(u)j

- a,) = a - an < E,

n>n which proves

the continuity

at u,,, and the proof

of the proposition

is complete.

We denote by SC-F the lower semicontinuous envelope (or relaxed function) the greatest lower semicontinuous function majorized by F (see [8,9]).

n of F, which is

at some PROPOSITION 2.2. Assume the function F: X + [0, +m] is not lower semicontinuous point u,, such that F(u,) < + 0~). Then, there exists a continuous function G: X --t R,, G(u,) = 0 such that the minimization problem (P) has no solutions.

t]ntl; (F(u) + G(4)

145

Lower semicontinuity

Proof. Since F is not lower semicontinuous at uO, we have F(Q) > SC-F(uO) 2 0. Applying proposition 2.1 to SC-F, we obtain a continuous function Y such that 0 < Y(U) I x-F(u) in X and Y(u,) = SC-F(u,) 2 Y(u) for all u E X. Let us define the function G by G(u) = -Y(U) + SC-F(u,) + d(u, u,). Clearly G is continuous, G(u) 2 d(u, u,,) z 0 and G(u,) = 0. On the other hand, for any u in X one has x-F(u)

+ G(u) = SC-F(U) - Y(u) = SC-F(Q)

This implies

(a

that u0 is a solution

+ SC-F(u,)

+ d(u, u,) L SC-F(u,)

+ G(u,).

to the problem rnn;(sc-F(u)

+ G(u)).

Actually, one can show that u0 is the unique solution. Let us prove that for the function G as above, the conclusion of the proposition holds. Suppose the contrary: let U be a solution to problem (P), then a well-known result (for instance [9, theorem 3.8]), asserts that min(F(u) + G(u)) = minsc-(F + G)(u) = min(.sc-F(u) + G(u)). Therefore, U is also a n solution to problem (p). Hence, ii = uO, thus F(Q) = SC-F(u,) reaching a contradiction. Remark 2.3. The previous proposition has an equivalent formulation. Let F: X + [0, +a] be a function such that problem (P) has a solution for every continuous function G : X + R. Then F is lower semicontinuous. THEOREM 2.4. Let F: X --) [0, +w] be nonlower semicontinuous at some point u0 such that F(u,) < + CD.Then, given any q > SC-F(Q) - inf F 2 0, there exists a continuous function G, , G,(u,) = 0 such that 0 I G, I q in Xand problem (P,), i.e. problem (P) where G is replaced by G,, admits no solution.

Proof. The function G, defined by G, = G A q with G given by proposition 2.2, satisfies the requirements of the theorem. In fact, if on the contrary there is a function U solution to problem (P,), then U is also a solution to problem (p,), i.e. problem (p) with G = G, . Therefore, since 0 I G, 5 G and G(u,) = 0 we obtain SC-F(U) + G,(u) I SC-F(u,) + G(u,) = SC-F(u,) < q + inf F = q + inf x-F % q + SC-F(U). Then G,(u) < q, i.e. G,(u) = G(ti), by recalling that u0 is the unique solution to problem (P), we conclude that k = u,,. Consequently SC-F(u,) = F(u,), which is a contradiction because u0 is a point where F is not lower semicontinuous. n Remark 2.5. That problem (I’) admits the unique solution u,,, the point just where F is not l.s.c., was the main fact used in the proof of proposition 2.2 (a similar argument has been used in [3] in a very particular situation, namely for functions of integral type). It turns out that problem (p,) has at most one solution: uO. This property fails if 0 < q < SC-F(u,) - infF, by assuming u0 is not a minimum for SC-F, otherwise see remark 2.6 below. In fact, suppose there is a function G, such that 0 I G, 5 q in X. Then, inf(Sc-F + G1) 5 inf sc-F + q = inf F + q < SC-F(q). Thus, there exists U in X such that SC-F(U) + G,(C) < SC-F(u,) I SC-F(u,) + G,(u,), i.e. u,, is not a solution to problem (p,).

F. FLORES

146

Remark 2.6. Assume

in the previous

6 = (G E C’(X,

theorem,

[0, +a[)

that u0 is a minimum

: G(u,)

= 0 problem

for SC-F in X. Then, the set

(P) admits

no solution]

is dense in the set of those functions in C’(X, [0, +a~[), vanishing at uo. Indeed, let G L 0, G(u,) = 0 be in C’(X, [0, +co[) and let q > 0. Then F + G is not a lower semicontinuous function at uo. Applying theorem 2.4 to the function F + G, we get a continuous function Go, G,(u,) = 0 satisfying 0 I Go s q in X such that the minimization problem + G(u) + Go@))

tnn;(F(u) admits no solution, our assertion.

thus

G + Go E 6

and

0 I (G(u) + G,(u))

3.THECONCRETE

- G(u) 5 q.

This

proves

CASE

In this paragraph, as in those which follow, 1. (, ( a, * > stand for the Euclidean norm and the inner product in R", respectively. For a given convex function f, we denote by df(t) its subdifferential at r: it is the set of subgradients off at r (see [lo, Chapter I]). We shall need the following auxiliary lemmas. 3.1. Let 6: R + R be a convex function such that 4(-t) = 4(t) for t E R. Then: (a) qb(t) 2 6(O) v t E R, and +(t,) 5 $(tz) whenever 0 I t, I t,; (b) 4’(O) 5 0 s 4:(O), i.e. 0 E &$(O); (c) c$:(tJ 5 $L(t,) _= +:(tz) whenever t, < t2. Therefore, if 4’(to) = 0 for some to > 0, then 4(t) = 4(O) for t E [-to, to]. LEMMA

Proof. Since r~5is an even and convex function, we have 6(O) I +4(t) + &b(- t) = 4(t). The second part of (a) follows from the assertion above, since one has q5(tJ % cu$(O) + (1 - ol)$(tJ, where we have written t, = a0 + (1 - a)t, . It is well known that the right- and left-hand derivatives of any convex function exist everywhere. For t > 0 (t < 0), using (a) we obtain o

~

4(t) - 443

(>) 7

t

letting

t -+ +0 (t + -0),

part (b) is proved.

LEMMA 3.2. Let 4 be as in lemma

Part (c) is left to the reader.

3.1 and let f: R" + R be the function

n

defined

by f(x)

= +(1x1).

Then

Proof.

By proposition

4.2.1 and 5.1.1 of [lo], we have

<* E af(O ifff(T)

+ f *(r*)

= (<*, 0 iff

iff Ad) + +*(lC*I)= k*I I51 where the last equivalence

follows

and

from the definition

4dtl) + +*(lt*I) = (5*, 0 <5*, 0

=

k*I ItI,

of $*. This proves the lemma.

n

Lower

147

semicontinuity

The previous lemma continues to be valid for Banach duality inner product between the space and its dual one. LEMMA 3.3. Let (T: R + R be a measurable, E > 0, n E N, C > 0, there exists a function

-C

spaces.

Here ( -, *> stands

locally bounded and odd function. q~in C”([O, E]) such that

Then,

5 v(t) 5 C if t E [0, .s]

for the

given

(3.1)

p(t) = 0 if 0 5 t 5 6,) 6, 5 t 5 E p(t) = C if a2 5 t 5 6,

and

if S, 5 t 5 6,

p(t) = -C

(3.2)

for some 0 < 6, < 6, < 6, < b4 < a5 < a6 < E, and E

o(cp(r))r”-’ dr = 0. i

Proof. Let t+~ibe an odd and Cm-function (see Fig. 1)

in [--P/2,

sn/2] satisfying

-CIy/,(t)ICift~

VI(t) = 0 if -z

&” a(tyz(t))dt

properties

1

5 t I -CT,, 8, 5 t 5 G

5 t 5 -6,

for fixed 0 < 6, < 6, < 8, < P/2.

the following

-5,: [

VI(t) = C if -6,

(3.3)

0

Then,

and

VI(t) = -C

if 8, 5 t 5 6,

by putting

yz(t) = vl(t

- Y/2),

= j;c+(t

- $))

dt = j;E;20(y/,(t))dt

we have = 0

i0 since cr 0 t,~r is an odd ~(4 = ~~(0, that

function.

Similarly,

an easy computation

shows,

for the function

E a(q$r))r”-’

dr = 0,

0

which proves the lemma with aj = ;/c”/2

- J4_;, cS;+~= w

for i = 1,2,3.

n

of this paper, 1 < p < + co, h : R” -+ R is a continuous function such that: (h,) 40 = GM) f or some even function i; defined in R; (h,) there are positive constants 01~I CY~ such that ~~ilrl~ I h(c) I c~(lrl~ + 1). In what follows, we denote by h* and h ** the polar (or conjugate) and bipolar function of h, respectively. For their properties we refer to [lo]. We just recall one: h** is the largest convex function not larger than h. As a consequence of (h,) h** is also continuous. The set ~2 will be a bounded and open set of R” with smooth boundary X2. We notice that (h,) implies the existence of an even and convex function C#Jsuch that h**(c) = +(lrl) and h*(c) = +*(lrl) for all c E R”. Actually I$ = i;** (see [lo]). In the remainder

We have the first main result of this section.

148

F. FLORES

Fig. I.

THEOREM 3.4. Assume h**G) second

the function h be as above. If h is not convex, in the sense that in its f M,) for some to, then, there exists a function g: Q x R + R, continuous argument such that the following problem

(PI)

min I, E w;.P(n)

h(Vu(x))

dx +

n

IR

g(x, u(x)) h,

has no solution.

Proof. Set t, = Ito1 and let us fix x0 E Q and E > 0 such that B, 5 B(x,, E) CC Sz. Let 0 < 6, < 6, < 6, < 6, < E and let us consider a function v, in C”([O, E]), as in Fig. 2, satisfying if t E [0, c]

0 5 P(f) 5 d’(4)) v(t) = 6’(to),

$40 = 0

3

if 6, I t 5 6,

if 0 5 t 5 6,) 6, 5 t 5 E.

Here the function 4 is as before, that is, satisfies h**(t) = $([
Fig. 2.

= @‘(l&l) exists

Lower

149

semicontinuity

Since 4’(to) E &#J(Q, that is, &, E (134)~‘(+‘(t,)), by recalling that (a$)-’ = a4*, to E a$*($‘(&)), so that it is possible to choose a function o E L;",,(R) selection from the map s - 13$*(s) such that a(+‘(&,)) = t, : for the existence of such a measurable selection we refer to [ll], and by (if necessary) we obtain the selection desired. assumption (h2) it is in L,“,, , after a modification Thus, a(cp(t)) E a+*(&t)) or, equivalently, p(t) E d4(o(p(t))) t E [0, E]. We now consider the function u&) where u1 is the solution

for x E B(x, , E),

= Ul(lX - x01)

to the equation

From the definition of u0 and 0, it follows that u0 a(& Ix - x01)) for x E B,. We still denote by u0 Clearly u0 E W;*“(Q), and by the choice of &, and value t, in B(x,, 6,)\B(x,, 6,): a subset of 51 with h**(vu,(x))

E H$r,P(B,) and IVu,(x)( = la(p(lx - x,J))] = the function extended by zero outside B,. by noticing that u0 has a gradient taking the positive measure, we have

WhdxN dx.

dx <

i a

i

(3.4)

R

On the other hand, we know that ~(lx - x0/) E 134(0(&/x - x01))) for all x E B,. Then using lemma 3.2, we conclude that &lx - x,J)((x - x,)/Ix - x01) E #r**(Vu,(x)) a.e. x E B,. Calling p0 the Cm-function such that p;(r) = q(r), pa(e) = 0, and if &(x) = p&(x - x0() then PO E C,“(B,). We extend this function by zero outside B(x,, E), to obtain a function, still denoted by Do defined in Q. Taking into account that 0 E ah**(O) and the properties of (4, we obtain V&,(x) E ah**(Vu,(x)) a.e. x E Q. Therefore, h**(vu(x)) for any function we obtain

I h**(Vu,(x))

+ (V&(x>, Vu(x) - Vu,(x))

u E Wd*“(sZ). On integrating

for a.e. x E Q

(3.5) over Q, and by using

Green’s

(3.5) formula,

0 I ,R

h**(vu(x))

2

h**(vu,(x))

dx .i a

.i n

&%(x)(u(x)

- u,(x)) dx

for every function u E WdP”(sZ). Setting pi(x) = A&,(x), the last inequality solution to the relaxed problem min h**(Vu(x)) 11E w;.p(a) ,i n

(p;**) where g is defined

dx + .i cl

implies

that u0 is a

s(xt u(x)) d-&

by g(x, u) = P,(X)(U - u,(x)) +

Iu - WP,

(3.6)

which is continuous and strictly convex in U, so that u0 is the unique solution to problem (Pi”*). Let us prove that problem (PI), with g as above, does not admit any solution. Suppose the contrary, i.e. let u be a solution to problem (PI), since min PF* = inf P, (see [lo, theorem 3.7, Chapter Xl), u is also a solution to problem (Pf*). Therefore, u E u0 because of the uniqueness of solutions to problem (Pf*). It follows that ia h**(Vu,(x)) dx = Ia h(Vu,(x)) dx reaching a contradiction with (3.4). n

150

F.

FLORES

Remark 3.5. In theorem

5.1 of [7] a similar result was established, but here we had some difficulties in constructing continuous perturbations, since we do not allow the boundary datum to vary as occurs in [7], in other words, the space where we seek the minimum is fixed from the beginning. THEOREM 3.6. Let h: R --) R satisfy hypotheses (h,) and (h2). If h**(t,,) # h(t,) for some t,, R, continuous in its second argument such that the then there exists a function g: Q x R --f minimization problem

(P2)

min h(Au(x)) u t wp(n) II R

dx +

R&4x, a))

dx,

has no solution.

Proof. We assume to L 0, otherwise we consider -to. We proceed theorem 3.4. First, fix x0 E Sz and E > 0 such that B, 5 B(x,, E) CC Q. (a) Let us assume h**‘(tJ > 0. Since to E (ah**)-‘(h**‘(fO))

as in the proof

of

= ah*(h**‘(t”)),

and 0 E (ah**)-‘(O), we can take any measurable, locally bounded selection (T from the verifying a(h **‘(to)) = t, and o(O) = 0. After a modification map s u ah*(s) = (ah**)-‘(s) (if necessary), we can assume, because of the properties of h**, that CJ is an odd function. Consider the function v, given by lemma 3.3, with C = A**‘(@, and let U, be the solution to the Dirichlet problem

= o(dx - x01))

4, u()

=

0

(3.7)

aB,.

on

Then u,, E PVZV’(B,) f’ I&iPP(BE), and by virtue of (3.3) we obtain au,/& = 0 on dB,, thus u0 E W:*“(B,). By the definition of CJ, we have p(r) E &?**(a(@-))) r 6 [0, E], or equivalently rp(]x - x01) E ah**(@(lx - x0]))) x E B,, i.e. 9(1x - x0() E ah**(Au,(x)) x E B,. Setting Q(x) = ~((x - x,1) for x E B, and 3(x) = 0 for x E Q\B,, we have 6 E C,“(Q). We also extend u0 by zero outside B,, then u,, E PV:2P(fi) and by the choice of t, we have h**(Au,(x))

dx <

h(Au,(x))

i .n Recalling the properties for every 24E W,2,p(sZ) h**(Au(x)) On integrating

of h** and 0 we conclude

2 h**(Au,(x))

the last inequality

that Q(x) E ah**(Au,(x))

+ @(x)(A~(x) - AuO(x))

over Q and using Green’s

(!

h**(Au(x))

h**(Au,(x))

I

dx +

a

for a.e. x E 52. twice, we obtain

!

A@(x)(u(x) - u,(x)) dx

.Q

u E Wt*p(sZ). This shows that ug is a solution min h**(Au(x)) 11E w,z.poN <\ n

to the problem

1

7

(pz**>

formula

a.e. x E .Q. Hence,

0

I,

for every function

(3.8)

dx.

dx +

g(x9 u(x)) d%
Lower

where g is defined

151

semicontinuity

by (3.9)

g(x, u) = P,(X)(U - %(X)) + Iu - %l(x)V

and pi(x) = -A@(x). Obviously g is a continuous and strictly convex function in U. Therefore, u,, is the unique solution to problem (Pz*). It remains to prove that problem (PJ has no solution. If it is not so, any function u solution to problem (PJ is also a solution to problem (P;*) (see the proposition in the Appendix). Then u = u,; thus we obtain Ja h**(Au,(x)) dx = ia h(Au,(x)) dx, which is a contradiction to (3.8). (b) If h**‘(&,) = 0, we take any (for instance, radial) function ui E W,*P(B(O, E)) whose Laplacian takes values in l-t,, , to). Then, we define the function ua as follows: u,(x) = ui(x - x0) for x E B, = B(x,, E) and zero otherwise. Because of (a) and (c) of lemma 3.1, we have for every function u E IV:P”(Q), h**(Au(x)) 2 h**(Au,(x)) = h**(O) for a.e. x E Q. This implies that problem (P;*) with pi = 0 in (3.9), has the unique solution ~a. A similar reasoning as before allows us to conclude that the corresponding problem (P2) has no solution. W Remark 3.7. The function g in the two previous theorems can be chosen it suffices to consider g(x, U) = g(x, u) V 0 with g as above.

nonnegative.

Indeed,

Acknow/edgemenfs-The author wishes to thank the referee for his suggestions that improved the proof of lemma 3.2, making it shorter than the original one. He also gratefully acknowledges fruitful discussions with Professor G. Dal Maso during the preparation of this work. His suggestions have been greatly appreciated. REFERENCES 1. CELLINA A., On minima (1993). 2. CELLINA A., On minima

Analysis 20(4), 337-341

of a functional

of the gradient:

necessary

conditions,

Nonlinear

of a functional

of the gradient:

sufficient

conditions,

Nonlinear Analysis 20(4), 343-347

(1993). 3. MARCELLINI P., A relation between existence of minima for non convex integrals and uniqueness for non strictly convex integrals of the Calculus of Variations, in Mathematical Theories of Optimization, Lectures Notes in Mathematics, Vol. 979, pp. 216-231. Springer, Berlin (1983). problems, Communs partial difJ Eqns 14, 4. RABIER P. J., New existence results for some nonconvex optimization

699-740 (1989). for optimal control problems governed by 5. RAYMOND J. P., Existence theorems without convexity assumptions parabolic and elliptic systems, Appl. math. Optim. 26, 39-62 (1992). Ann. Inst. H. Poincurk, Analyse non LinPaire 6. TAHRAOUI R., Regularite de la solution d’un probleme variationnel,

9, 51-99 (1992). I. BALL J. M. & MURAT F., W’,p -Quasiconvexity

and variational

problems

for multiple

integrals,

J. funct. Analysis

58, 225-253 (1984). 8. BUTTAZZO G.,

Semicontinuity,

relaxation

and

integral

representation

in the Calculus

of Variations,

Pitman

Research Notes in Mathematics Series, Vol. 207. Longman, Harlow (1989). Boston (1992). 9. DAL MASO G., An introduction to r-Convergence. Birkhauser, Amsterdam (1976). 10. EKELAND I. & TEMAN R., Convex Analysis ond Variational Problems. North-Holland, 11. KURATOWSKI K. & RYLL-NARDZEWSKI C., A general theorem on selectors, Bull. Acad. pol. Sci. Ser. Sci. math. astr. phys. 13, 396-403 (1965). APPENDIX Here we are given a function h: Q x R + R such that (hi) for every [ E R h(. , () is C-measurable; (hi) for almost every x E s2 h(x, .) is continuous: (h ;) there are positive constants 01~, 01~ such that ol,l[lP I h(x, c) 5 olZlclp + p(x) for some nonnegative in L’(a).

function

/3

152

F. FLORES

Let F: w*,p(n)

+ [0, +m] be the function

defined

Ii n

F(U) =

by

if u E W,Z*p(n)

W, Au(x)) dw

+oO We also consider

H defined

the function

otherwise.

as follows

i

h**(x

H(u) =

.R

Au(x)) dx

if u E W,2,p(Q)

’

+oO where h**(x, c) is the bipolar

of the function sc_(w

otherwise,

h(x, .). From proposition

-WZ.P)F(U)

= SC,&0

= min

1.3.5 and remark

1.3.6of [8], it follows that

- WZ,P)F(U)

lim inf F(u,):

t h-+CC

,

uh 2 u in JV’,’

I

where SC-(o - W’,“)F, SC,~(W - WZ,p)F denote the lower semicontinuous envelope of F for the weak topology of W2,p and for its sequential version, respectively (see [8]). We are now in a position to state the following theorem.

THEOREM.

Proof.

sc&(o

Clearly

- WZ,p)F=

H.

H is 1.s.c. in the strong

H(u) 5 X&(W

topology

~ Wz,p)F(u)

of W’~“(a).

= min

So that,

lim inf F(u,,): u,, - u in W*,p h-f+-

vu E WQ(Q).

In order to prove that the last inequality is, in fact, an equality, we shall prove the following: E > 0 there exists a sequence (u,) in W2,p such that uh - u in W*,p(Q) and

given any u E W2.p and

5 H(u) + E.

lim inf F(u,) h-+0=

(1)

If u $ Wi,p(Q), the inequality holds trivially, so long as we may assume u E W,z,p(s2), Au being in Lp. Proposition 1.2 and 2.3 of Chapter IX of [lo] (or theorem 2.6.4 of [8]), imply the existence of a sequence (u,), uh E W2~p(Q) n Weep such that: Au, - Au in Lp(Q) and

i h**(x , Au(x))&

=

lim

c h(x, Au,(x))

b.

(2)

Clearly uh - u in PV2,P(!2). Let us fix a compact subset K of Q, setting 6 = dist(K, an) > 0, we consider the sets A,, i= 1, . . . . v, defined as follows: A, = Int K, A, = {x E R” : dist(x, K) < id/v] and functions g, E C,“(A,) such that 0 5 cp, 5 I in R”, rp, = 1 in A,_, and ID”cp,l zs M(v/~)‘“’ (a1 5 2, where M is a fixed positive constant. Define wi h = cp,u, + (1 - lo,)u, clearly w,,h E wtXP(C2) for every i = 1, . , v, and v h E N and w,,~ - u in WzSP for every i. 0; the other hand,

I

.n

h(x, Aw,,,&)) dx =

5

W, Au,,(x)) dx +

/)

W, Au,(x))

.n

dx+i

,i nw,

,, R\K

h(x, Au(x)) ti

h(x, Au(x)) du +

+

hOc,Aw,,,W dx

h(x, A~i,,,(x))

k.

(3)

Lower Let us estimate

the last integral,

Ir,h 5

W,

VP+ c4 ;

Aw,,,)

01

where c, ;

by the properties

Ivu, - VulPdx +

,.%\a,-,

Since Au, - AU in Lp, ilAuhll is bounded dx =

1

depending

P(x) d-c only on n, M and p. Then,

in Lp, say by c. Returning

to (3), we obtain

for some i,,

min

By a standard diagonalization procedure, we have the existence By (2) and recalling that uh - u in IV*.“, we get lim inf h(x, Aw,) ti i h-+,,R Choosing

(hi)

+ c2

. A,W,-,

i = 1, . . , 4, are positive constants

h(x, Awi,J

of 9, and assumption

IAz#dw

5 c,

153

semicontinuity

I

, I

h**(x, AU) ti +

of a sequence

(w,) converging

h(x, AU) du + y

+ :

weakly to u in ~+‘t,~(fi).

lIA#’

+ 1 llplll. V

.R

K and v such that _I

I

h(x, Au) du < f

and

ccl + c,llA#’

+ IlPil,) < ;>

<,RM we conclude

that lim inf F(w,) /I-+E?

which proves sc&(w As a consequence

- WzSp)F(u) = H(U).

we have the following

5 H(U) + E,

W

proposition

PROPOSITION. s&(w - W’,“)(F + G) = sc&,(w - WZzp)F + G for every sequentially G: W2,p(fi) + R. In particular we infer, for G as before, that

Remark. If h depends

only on <, it is not so hard to prove directly min h(Au(x)) dx = min UE F++“(n) n UE u&n,

In

that

h**(Au(x)) dx = meas(Q)h

weakly

continuous

function