Local minima of nonconvex problems

LOCAL

MINIMA

OF NONCONVEX

PROBLEMS

ELVIRA MASCOLO Instituto di Matematica “U. Dini”, Universita’ di Firenze, Vle Morgagni 67, 50139, Firenze, Italy (Received Key

29 July 1988; in revised form

9 February

1989; received for publication

6 June 1989)

Local and absolute minimum, convexity, lower semicontinuity,

words and phrases:

relaxation

methods. INTRODUCTION IN THIS

paper we discuss local minima of the functional F(u) =

q~(Du)d_x

-

[Gjklx

-

I’G

I’,,godS,

u E H’**(G),

(0.1)

where G c RN is a bounded open set with smooth boundary aG and dS is the (n - 1) dimensional area element on aG. We shall assume that cp is a Lipschitz function satisfying: C,lP12 - c* 5 $0)

5 c,(l + IPI*)*

p E RN,

(0.2)

for some c, L 0 and f E L*(G), g E H’**(G) satisfy I’/(x)&

+ iacgdS

= 0.

(0.3)

The notion of local minimum depends on the topology assigned to the test function, one feature of nonconvex functionals which distinguishes them from the more commonly encountered convex ones. One interpretation of this was offered by Ericksen [2] in the one dimensional case, where he considered the equilibria of bars under finite deformations. As we shall remind the reader shortly any local minima of his in energy norm is an absolute minimum, but there may be serval local minima, for example in C’ norm which are not absolute. Conceivably, any energy minimum may be found by duality or relaxation. The scope of this note is to examine the extent to which this property may be generalized to higher dimensions. We show that if z.+,is a local minimum in H’ of (O.l), then under suitable hypotheses on (D and f h,(x) E [RN - K a.e. in G, where K = @ E [RN: p**(p)

<

cp(p))

and up**is the lower convex envelope of cp, and

u,, is an absolute minimum of F. In addition, we prove an existence statement for F, which is not obvious since F is not below semicontinuous. To be precise, we introduce the space H’**(G) = HLPu, 1 I 01< + Q), of the distributions u on G for which the norm

I14.a = (~ya

+ IDulvq”

In the following we denote by H’ the space H’**. 593

< +co.

594

E.

MASCOLO

We say that u,, is a local minimum of F provided u O,G = (meas G)-’ lG u,, du = 0 and for some e > 0 V u E Hi, uo = 0, JIU”- UIIi_ < E. F(u,) 5 Z=(u) (0.4) As

motivation

for what follows, we briefly return to Ericksen’s problem:

(0.5) ~(0 dx - a(u(1) - u(o))

F(u) = I

where I = [0,11, 9 is a nonconvex smooth function and o is a real constant. Existence may be ascertained by applying Jensen’s inequality to 9**, for which purpose we introduce the problem P*(u)

= Min[P*(u),

P*(u) = iI

&(]O,

ll),jruh= O]

cp**(u')dx - a(u(1) - u(0)).

(0.6)

For any u P*(u)

> 9**(u(l)

- u(0)) - a(u(1) - u(0)).

If to is the minimum of the real function h(f) = 9**(f) - at, t E R the function u0 with u;(x) = to having average zero, is a solution of (0.6). On the other hand t, must obey the condition 9,r*(to) = Q (0.7) in order that u0 satisfy the Euler-Lagrange equation of (0.6). Suppose for simplicity that 9 # 9** in the interval K = (t, , tJ and let b be the number such that 9:*(t,) = 9t(t,) = 9r(t*) = 9:*(f2) = a so p:*(t) = b in (tl, t2). It is easy to check that if o # a, then (0.7) has three solutions, say (pi, cr2, ax, with a3 d (t, , tz). The function with uh = (Yeis a solution of (0.5). If o = b, then any function a with ii = t,xr, +

t,xr,

where xF is the characteristic function of F C [0,11, I = I, U I*, I, fl Z, = 4 and {,a(~) dw = 0, is a solution of (0.5). That any local minima of (0.5) in the sense of (0.4) is an absolute minimum is an observation made in [5]. We extend this result to higher dimension. Let us point out that the results rely in a crucial way on the fact that we deal with local minima in integral norms. Indeed, let us turn to one-dimensional problem (0.5). Let cr # or and as before (it, az, a3 the solutions of (0.7). If a3 d (tl, tz), (Pan < 0 and (Y, E (tl, t2) cpt,(cr,) > 0, we have that the function u,,(x) = (Y,(x - l/2) is a local minimum of F in the C’ topology which is not an absolute minimum. In the second part, we state an existence theorem for functional (0.1) in H’. The proof relies on the methods of [g-lo]. We construct a special minimizing sequence weakly convergent to a minimum of relaxed problem. The methods of variational inequalities are then employed to

595

Local minima of nonconvex problems

conclude

that

this sequence

converges

strongly

in order to exhibit a solution of nonconvex

problem. 1. EQUIVALENCE

Let G c functional

RN,

p,

OF LOCAL AND ABSOLUTE

f, and g as in the Introduction, F(u) = [$Du)

MINIMA

satisfying (0.2) and (0.3). Consider the

- iGfub - iacgudS.’

In the sequel we refer to a local minimum of F in H’, defined as a function u0 with uOG= 0 and for some E > 0 for each u E H’, uG = 0, IIu - u,,lll,Z I E. F(u,) 5 F(u) We say that u is an absolute minimum if uo = 0 and F(u) = Min(F(u), u E H’, uG = 0).

Set K = Ip E lRN: p**(P) c (p(P)), where v,** denote the lower convex envelope of cp. K is an open set, since q and cp** are continuous functions. Let us assume: (i) The set K is a bounded convex subset of RN and (p is a C’ function in RN - K. Moreover cp** is affine in K, i.e. there exist N + 1 constants, m;, q such that for p E K v**(p)

(1.3)

= c mipi + 4;

(ii) f E Co(G) and meas(x E G : f(x) = 0) = 0. In the sequel we shall utilize the following proposition, theorem 1.1 of [9] and the results of Chapter V of [6].

which may be derived easily from

PROPOSITION 1.1. Let K, Q be bounded subsets of U?, N 2 2. If w E H’*“(G) such that Dw(x) E K a.e. x E 51, then there exists at least one solution u in H’*“(G) of the problem

DU(X) E aK, u = w,

a.e. in a, on

(1.4)

an

In particular, the maximum and the minimum element of the set S S = [u E H’*“(G),

Du(x) E R a.e. in Sz, u = w on aa),

i.e. the functions u+(x) = suplu(x),

x E 8,

u_(x) = inf(u(x), x E S),

are two solutions of (1.4). We also use the Poincare inequality: for each u E A!Z’~“(G)

IIU- b3111,u(G) 5 C, IlmlrqG,% where the constant C, depends only on G, CY,and N. Our main theorem may be regarded in a sense as the generalization property of convex functionals. ‘For a function h(x), we denote by Dh(x) the vector gradient Dh = (D,h, . . . . D,h).

(1.5) to F of a well known

Dih = ah/ax,.

E. blASCOL0

596

THEOREM1.2. Assume (0.2)-(0.3)

and (i)-(ii). Let u,, E H,>c”(G) be a local minimum of F. Then Du,(s) E I?” - K

a.e. in G.

(1.6)

Moreover uO is an absolute minimum of F. Proof. We first prove (I .6). Owing to Rademacher’s to points x0 E G, where u. is differentiable.

theorem,

we may confine

our attention

Suppose Duo(xo) E K, From lemma 4 in Appendix Du,(x,)

f@o) > 0.

(1.7)

2 of [6], there exist two functions

= DY+(x,)

Y+, Y_ E Cd(G)

= DY-(x0)

uo(xo) = Y+(xo) = Y-(x0) Y-(x)

such that

< u,(x) < Y+(x)

(1.8) for some 6 > 0

x E B, b-o) - hoI

where B&(x0) = BB = {x E RN: Ix - x01 < 6). Further, since u. E H&-(G), there exists L > 0 such that sup[lDY+(x)( By restricting

+ ]DY_(x)]

to r < 6, we suppose

(1.9)

that: and

f(X) > 0

+ IDu~(.Y)~, x E B6) 5 L.

DY-(x)

E K,

(1.10)

x~Br.,

and meas B, c e(4(2L2 + kf2]nC2)-‘, where M = sup1 Ix - y], (x, y) E IQ and C2 is the constant is an integer such that nC2 > 1. We consider the problem

Dx(x)

E

aK,

proposition

X+(X) = sup(v(x),

(1.5) for CY= 2 and n

in inequality

a.e. .Y E B,, (1.12) on dB,.

x = Y-, In view of (1. 10)2, by applying

(1.11)

1.1, the function

defined

as

Du(x) E R a.e. in B,, u = Y_ on as,),

is a solution of (1.12). Let now < E C,“(B,) with supp r c B,,2, 0 I t(x) 5 1 and 0 such that D(Y_

+ T u. in A

E

K,

x E B,.

+ T uo(xo) and x+ = Y_ < u. on aB,, there

and

x+ =

24,onbl.

Local minima

of nonconvex

591

problems

Let us consider the function of H’*“(G) defined by XEA

x+(x) W)

x(x) =

(1.13)

XEG-A

and X(X) = x(x) - KG. We show that: (1.14) First, we observe that

llm - mxZ(,4, 5 wx+ - ~~-l&.A, + WeSince Ox+ E aK and DY_

- mx~(,4,.

E K in B,, we have

sup[]Dx+(x) - DY-(x)1,

x E B,l 5 M

then I M2 meas B,.

IlOx+ - DY_lli~,~)

In addition, from (1.9) we get - Duoll~zcAj s 2L2 meas B,.

llDY_

Finally, from (1.11): < 2[M2 + 2L2] meas B, I e(2nC2)-‘.

IID/? - Du,t]&, Moreover,

inequality (1.5) gives 11x- u,]]?+G) 5

C211Dx

-

D~,ll&~

5

d24-‘,

which implies (1.14). Compare now F(u,) and F(ji), for which we remind ourselves that for each p E Pv and the equality holds for p E k‘.

P(P) 2 c miPi + 4 Moreover,

from (0.3), FQ) = F(x). Therefore F(u,) - F(x) =

r:c Now,

s B,

@(Due) - &Dx)) dx miDi(U,

s ‘9,

- X) dx -

f(kJ - x) k 5 B, f(kl s Br

- x) dir.

since x = u0 on aB,, by divergence theorem, we get

s s

C miDi(U,

- X) dw = 0.

Lb

Moreover, since x > u0 and f > 0 in A, -

f(u0 - x) dx > 0.

A

In conclusion, F(u,) > Fk), which is a contradiction to the fact that ue is a local minimum. Then, for each x0 in whichf(x,) > 0, we have Du,(x,,) E RN - K. In analogous way, by using

E. MASCOLO

598

Y+ instead of Y_ , we may show that there are no x0, where u0 is differentiable D%l(-%) E K

and

f(x0) < 0.

Then we may conclude that for each x in which u,, is differentiable (1.6) holds. By the Euler-Lagrange equation we have for u E H’ and uG = 0 n g(u - u,) dS. (Pp(D4P(n - 4)) ~ = f(n - 4) +

(1.15)

ac

G

G

and

Moreover, since, by (1.6), @*(Du(x)) = (p,(Du(x)) a.e. in G, we get yl**(DU,) d_X1

fJI**(Du) d.x IG

.rG

iG

VJ&JD(~

- %I) dx.

(1.16)

By combining (1.15) and (1.16) we obtain F(U,) = P*(U,)

5 P*(u)

I F(u),

which completes the proof. The preceding theorem was formulated under the assumption that the local minimum u,, E H,i;,“(G). For some problems important in applications we do not know if the local minimum satisfies the latter requirement. Therefore, now we prove an analogous statement for a different class of local minima. Suppose that a, satisfies (1.17) &IQ - cz 5 V(P) -( cslPla + c4, with 2 < CY< + co, and iet f E L”(G) and g E H’*“(G) satisfy

5G

“f-b+

gdS = 0.

Define the functional as in (1.1). We say that u. E If”” u eo = 0, for some & > 0 Wo)

5

(1.18)

s ac is a local minimum of F if, provided

for each u E Hlsa’, uo = 0, ((u - uOI(l,a < E.

F(u)

The following theorem holds. THEOREM

1.3. Let cp,f and g satisfy (1.17)-( 1.18) and suppose

IVp(P)l5 cs(l + lPl”-‘).

for p E RN - K.

Moreover, assume (i)-(ii). Let u. be a local minimum of F in Zfl*OL,with cu > N. Then u. is an absolute minimum i.e. u. satisfies F(u,) =

Min(F(u), u E HlsU, uG = 01.

Proof. Since most of the proof is a repetition of the proof of theorem 1.2 we will only indicate some particular points where a new argument is needed. First, we observe that, since u. E H’*“(G), with (Y> N, from Sobolev embedding theorem, u. is Holder continuous. Moreover, no is differentiable a.e. in G (see for example [l 11).

Local minima

Let x0 E G where u,, is differentiable

of nonconvex

599

problems

and suppose and

Du,(x) E K

(1.19)

J-(&J > 0.

Consider the functions Y_ E Cd(G) satisfying (1.8). Let C, be the constant in equality (1.5) and j E N with j > 1 and jC, > 1. In accordance with the continuity property of integrals, there exists oE > 0 such that, for each measurable subset Go c G with meas Go < os we have lDuOlu ti < .s(42”-‘jC,)-‘. .rG Let P c 6 such that

f(x) > 0

and

DY_(x) E K

for x E B,-,

and meas B, < min[c[42*(2”-‘L”

+ M”JjC”]-‘,

a,]

where L > sup(]DY_(x)], x E BJ. By considering the solution x+ of the problem (1.12), we may construct x as in (1.13) and X=x-xc*

Let us estimate [IX- u,,]],,a. First we have ]]Dx - DY _ l];~(~, I M” meas Bi . Moreover, since meas Bi

c

aE,

]]DY_ - Du,]],“~(~, I 2*-‘L” meas Bi + &(4jC,)-l, we deduce ]]I@ - Du,])~u(,, I 2”[M* + 2”-‘L”] meas B,,

+ &(4jC,)-’ = E(2jCa)-l.

Finally, from Poincare inequality (1.5), we get

By proceeding as in theorem 1.1, we have that IQ) < F(u,), so there are no x0, where u,, is differentiable, in which (1.19) holds. Moreover, by considering Y, , instead of Y_ , we show that Du,(x) E IR” - K a.e. in G. By using the Euler-Lagrange equation and the convexity of rp in [R” - K, we may conclude as above that u,, is an absolute minimum. 2. EXISTENCE

THEOREM

Let us formulate in this section an existence theorem for nonconvex problems. The proofs given below reflect the differences between this problem and the Dirichlet problems. Let G, cp, f and G as in the previous section and consider the problem F(U) = Min(F(v), v E H’, uo = 0],

(P)

where F(V) = jo @v) ti - jo fu dx - jaGgu dS. Assume that 9 E C’(RN) and ForpERN

ClP125 9(P).

(2.1)

E. bfASCOL0

600

The set K = (p E RV, 9**(p) < 9(p)] is bounded and 9 is a strictly convex function in R“’ - K, and p E IR.v - K. I9JP)I 5 c(1 + IpI), (2.2) Moreover, (V,(P)

- vp(q))(p

p, q E IR” - K.

- 4) 2 4P - q12

Further there exists M E R such that in K.

9**(p) = M

(2.3)

9(p) = h(lpJ), p E IR”, where h is a C’ function in R such that h(t) > Let h(l,) = min(h(r), t E R+J, suppose that h is strictly convex for t > t,. It is easy to check that Set, for example,

c,t2 + c2, t E R.

IPI 5 to

9**(p) = We) for

and

9**(p) = NIpI) for

IPI> lo9

then the function 9 satisfies (2.3). The following existence theorem holds. THEOREM2.1. Let 9, f, g satisfy (0.2), (0.3) and assume (2.1)-(2.3). Then, there exists u E H’, having uG = 0, such that F(U) = MinfF(v), v E H’, vq = 0). In the sequel, we suppose that 9**(p) = 0 in K; indeed in the case M # 0 it is sufficient tc consider 9 - M. Before passing to the proof of theorem 2.1, let us establish some preliminary results. Let Ft* be the relaxed functional of F, F**(v)

9**(Dv) dX -

= G

_iGfv

du

-

j,

gv dS,

we consider the relaxed problem of (P), p*(u)

= Min(p*(v),

v E H’, vG = 0).

(p**)

In view of well-known relaxation results (see for example [ 1, 71) inf (P) = inf(P**). Let us define, for L > 0 Y, = (v E H’*m, llDVllm 5 L, UC = 01,

and consider the problem p*(u)

= Min(Fr*(v),

Since F** is a semicontinuous functional, at least one solution u of (Pt*).

v fz L.,].

(pi!*)

by using the direct methods arguments, there exists

PROPOSITION2.2. Let 9, f, and g satisfy (0.2)-(0.3) and assume (2.1)-(2.3). Then, there exists with L 2 Lo, has a solution u satisfying

Lo such that (Pt*),

Du(x) E RN - K,

a.e. in G.

(2.4)

Local minima

of nonconvex

601

problems

Proof. First, we assume that f E Co(G) and meas{x E G :f(x) = 0) = 0. The arguments are similar to those developed in [g, lo]. Let u be a solution of (Pt*). Let x0 E G, where u is differentiable, be such that Du(xo) E K and (2.5) Du(xo)I < L.

Let Y_ E C,‘(G), satisfying (1.8). Let f(xo> > 0 and 6 > 0 be such that for x E B6 DY-(x)

E

IDY-(x)1

K,

Let v E Cr(RN), supp rl C B6, 0 5 V(X) I 1 and a D(Y_ + V)(X) E K,

and

< L

Imy-

f(x)

> 0.

(2.6)

= 1. Moreover let r > 0 be such that

+ v)(x)1

< L.

x E B,.

From the properties of Y_ , there exists an open subset A c B6 such that (Y_ + TV)(X)< u(x), XEA andY_ + rq = uon&4. Let us define w(x) = max((Y_ + rq)(x), u(x)), and i+ = w - wo. Compare p*(u) and Ft*(rv), for which we observe that p E iRN

(P**(P) 1 0,

Moreover (0.3) implies c;Y*(ii)) = p*(w).

and

v**(P)= 0

for p E K.

Thus from (2.6), we have

F**(u) - F**(w) = - r f(u - w)dx > 0, JB i.e. Fc*(iV) < F**(U). The latter inequality is in contradiction to the fact that u is a solution of (Pt*). By considering Y+ instead of Y_ , we show that there are no x0 in which u is differentiable, f(xo) < 0 and where (2.5) holds. Now, since K is a bounded subset, let Lo > 0 be such that B(0, Lo) > K. We observe explicitly that Lo depends only on K. Thus, if u is a solution of (Pt*), with L 1 Lo, necessarily (2.4) holds. To complete the proof, consider a sequence fj of polynomials satisfying (0.3), which converges to f in L’(G). Consider the sequence of functionals:

Since each fi E Co(G) and meas(x E G :fj(x) = 01, from the first part of the proof, there exists Uj such that v*(Uj)

= Min[y*(u),

u E Y,),

LrL,,

(2.7)

and OUj(X) E mN - K,

a.e. in G.

(2.8)

In view of the coerciveness assumption, the norms of DUj are bounded in L2 and then there exists a subsequence of Uj, which we still denote by Uj, converging weakly in Zf’ to a function U. Moreover, Uj converges uniformly to u and u E Y,. Since, for each u E H’ lim y*(u) by using the semicontinuity p*(U)

= p*(u),

of F** and (2.7), we obtain

I limp*(Uj)

I lim y*(u)

= F**(v),

for each u E Y,.

E. hlASCOL0

602

The latter inequality implies that u is a solution of (P,**). We like to show that Ruj converges to DU a.e. in G, and thus, in view of (2.8), Du(x) E Rv - K a.e. in G. Since Uj and u are solutions respectively of (2.7) and (Pt*), the following inequalities hold

-

c

co,**(DUj)D(U

Uj) dX 1 rj(U -

J

Uj),

(2.9)

1

I

pz*(D,)D(Uj - U) dx 1 T(u~ - u),

where T and q are the following linear functionals n

I

w dS,

,dG 1

I

T(u) =

fududx

+

for u E L*(G). Set

s

gv dS,

aG

,G 7 Ij

=

(~~*(OUj)

-

co,**(OU))O(Uj

-

je

U) dx,

N.

I

By taking in account the convexity of rp**, it follows 4 L 0, for each j. In addition, since Uj -+ in L*, the inequalities (2.9), implies 0 I lim Let A = (X E G: D&c)

lim(q -

Zj 5

T)(U

-

Uj)

=

u

(2.10)

0.

E K). Then we get c

3

ID(U

-

Uj)l*

dU I

Ij,

G-A

and then the sequence Duj converges to Du in (L*(G - A))N. On the other hand, since

!;

(ql**(Duj)

- q7**(Du)) d_Y_(

s

~~*(DUj)O(Uj

-

U) dr

I

~(Uj

-

U),

G

G

we obtain $7**(Du) du.

p**(Duj) dX =

lim G

Moreover (p**(DU) -

p**(DUj))

dX

A

((p**(DU)

5 I

-

p**(DUj))

dX

IS G 0 +

(q**(DU)

-

~**(DUj))~

9

I! G-A

and then in particular we have _) lim

~**(OUj)

!A

du

fp**(Du)

= 4A

dx.

(2.11)

603

Local minima of nonconvex problems

By taking in account (2.1) and (2.3), (2.11) implies lim 1 lDUj12du I 0. ,A Consequently Duj converges to 0 in (L2(A))N. Thus, it follows that DUj converges in (L2(G))” to the function w such that w = Du in G - A and w = 0 in A. Since Ui converges weakly in H’ to u, we get w = Du. After a selection of a subsequence, we may conclude that DUj converges to Du a.e. in G. From (2.8) it follows that meas A = 0 and u satisfies (2.5). The proof is complete. Proof of theorem 2.1. Set j E N with j > L,. Let Uj be the solution of F**(Uj) = Min(FC*(u), u E 5,

(P;*)

such that a.e. in G.

DUj(X) E RN - K,

(2.12)

We observe that (2.12) implies Fc*(Uj)

=

(2.13)

F(Uj).

First, we prove that Uj is a minimizing sequence of (P**). Consider the decreasing sequence (Fc*(uj)), we have lim Fc*(Uj) i

=

inf

P*(Uj)

L

inf(P**).

In addition, since we may approximate each v E H’ with functions average, we get Fe*(U) L inf(P;*) = infFC*(Uj), and then lim Fc*(uj) = inf(P**).

in HI*-,

having zero

(2.14)

From the assumption (0.2), the norms of DUj are bounded in L2 and thus there exists a subsequence, which still we denote by Uj, converging weakly in H’ and strongly in L2 to a function u. The semicontinuity of Fc* and (2.4) implies that u is a solution of (P**). IfuEH’*-, there exists k > L, such that [[Dull_, < K i.e. u E Y,. We derive that u is also a solution of (Pz*). In view of (2.13), we obtain inf(P**) = p*(u)

= F**(uJ

= F(u,) L inf(P),

since (P) and (P**) have the same infimum, we may deduce that F(uk) = inf(P) i.e. uk is the required solution of (P). Let u $H’*“. We like to show that there exists a subsequence of Uj which converges to u in H’. To this end, let uk be a sequence in Hi*” with vk,o = 0 which converges to u in H’. Let jk an increasing sequence of integers, such that llDUkllm5 j, i.e. vk E qk. Consider Uk = the solution of (P$*) verifying (2.12). Since the following inequality holds ujk

vkj

(o,**(DU,)D(U,

-

5G where T is defined as in proposition

2.2.

U,)

dx

2

T(t,k

-

Uk).

E

Ljk

(2.15)

604

E. hlASCOL0

Moreover,

since u is a solution of (P**), we have

I I

co,“*(Du)D(uk

- u) dx =

qu, -

(2.16)

u).

.G

As before. set Ik =

(co,**(Duk) - (Dp**(DU))D(& - u) dx L 0.

c

Inequality (2.15) rewrites 1

7 $7;*(Duk)D(uk

q$*(Du)D(u

- u) dx +

- u,)dx

2 T(Uk - u,).

LI

L\

Therefore,

taking in account (2.16), we obtain II Zk = T(U - u/J co,**(DU,)D(U - ?Jk)d_X. ! From (2.2), since the norms in Lz of Du, are equibounded there follows the equiboundness the sequence cp**(Du,). Consequently, we get I, -+ 0, asj + 00. Let A = (x E G: Du(x) E K). By proceeding as in the previous proof we have

which implies that Du, converges to Du a.e. in (L2(G - A))“. To complete the proof, we verify that 1 rp**(Du,) dx = p**(Du) dx. lim sG IG Indeed, q$*(Du,)D(u,

(v)**(DuJ - p**(Du& dx I

of

(2.17)

- uk) dx 5 7j(u, - uk),

G

G

as k -* 00, since uk converges to u in H’, we obtain lim

o)**(DUk) dx I lim I‘G

i

v**(Du) dx.

p**(Dur) dx = sG

,G

(2.17) follows from the semicontinuity of JG rp**(Du)dx in the weak topology particular, from (2.17) and taking in account (2.5), it follows lim

of H’. In

fJI**(DUk)dX = 0, sA

Du, converges to 0 in (L2(A))N. Thus, by proceeding as in the last part of proposition 2.2, we may deduce that there exists a subsequence of Du, which converges to Du a.e. in G. It turns out that Du(x) E RN - K a.e. in G and then so

p*(u)

= F(u).

The relaxation equality shows that u is a solution of (P).

Local minima

of nonconvex

Acknowledgemenrs-The work in this paper was carried I wish to thank D. Kinderlehrer for helpful discussions

problems

out while the author was visiting the University and valuable comments on the subject.

605

of Minnesota.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

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