manuscripta math. 51, I - 28 (1985) manuscripta mathematica �9 Springer-Verlag 1985

APPROXIMATION OF QUASICONVEX FUNCTIONS, AND

LOWER SEMICONTINUITY OF MULTIPLE INTEGRALS

Paolo Marcellini

We study semicontinuity of multiple integrals ~ ~f~,u,Du)dx~

where the vector-valued function u is defined for xE ~C N

with values in ~ . The function f(x,s,~) is assumed to be

Carath@odory and quasiconvex in Morrey's sense. We give con-

ditions on the growth of f that guarantee the sequential low-

er semicontinuity of the given integral in the weak topology 1 N

of the Sobolev space H 'P(~;~ ). The proofs are based on some

approximation results for f. In particular we can approximate

f by a nondecreasing sequence of quasiconvex functions, each

of them being convex and independent of (x,s) for large val-

ues of ~. In the special polyconvex case, for example if n:

N and f(Du) is equal to a convex function of the Jacobian

detDu, then we obtain semicontinuity in the weak topology of 1,p(~ n

H ;~ ) for small p, in particular for some p smaller

than n.

i. Introduction

Let us consider a function f(x,s, ~) defined for x in a bound-

ed open set ~ of ~ n, sE ~ N and ~ ~ nN. We assume that f is a

Carath~odory function, i.e., it is measurable with respect to x and

continuous with respect to (s,g), and satisfies the growth conditions

(1.1) - c ~ l ~ l r - C 2 1 s l t - C ( x ) <_ f ( x , s , g ) _< g (x , s ) (1 + I<lP).

Here CI,C 2 >0; C3E L~ (n); g >_ 0 is a Carath4odory function (no

growth conditions are required for g). For the exponents we assume:

p>l; I <r <p (r =I if p = i) and l<t <np/(n-p) (t_> 1 if p>_n).

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Moreover, if C2~ O, we assume also that the boundary ~ is Lipschitz

continuous.

Finally we assume that f is quasiconvex with respect to { in

Morrey's sense ([25]; [26], section &.&):

, v l, N) (1.2) I~f(x,s [ + Dr >_ Ig If(x,s, ~) r o =(~ ;~ ,

and also for every r EHlo,P(~;~ N), by mean of (i.i).

In section ~ we prove the following result:

THEOREM l.l - Let f(x,s,~) be e Carath6odory function, quasi-

convex with respect to ~, and satisfying the @rowth condition (I.I).

Then the integral

i~3) I f(x,u(x),Du(x)) dx

is sequentially lower

Hl,P(~;~{ N).

semicontinuous in the weak topology of

Theorem i.I improves the analogous result by Morrey [25] ,

[26] and by Meyers [2& ], who assume a type of uniform continuity

of f with respect to its arguments, and a recent result by Acerbi

and Fusco [2], obtained assuming slightly more restrictive growth

conditions.

We recall that either if N : 1 or n - i, then f is quasicon-

vex if and only if f is convex; while, if both n and N are greater

than one, then quasiconvexity is a more general condition than con-

vexity (for properties of quasiconvex functions we refer to [26] ,

[5], [8], [20]; see also our section 5). Therefore it seems not pos-

sible to reduce theorem I.i to the semicontinuity results known in

convex case (see, i.e., 5errin [31], De Giorgi [i0], and more re-

centl~ Ekeland and Temam [15], chap. 8, theorem 2.1, and Eisen

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[13]).

The proof of theorem 1.1, d i f f e r e n t from t h a t of [25] , [ 2 ~ ] ,

[2 ], bu t s i m i l a r l y to o ther c l a s s i c a l s e m i c o n t i n u i t y r e s u l t s , is b a s -

ed on the p o s s i b i l i t y to a p p r o x i m a t e f by a n o n d e c r e a s i n g sequence

of f u n c t i o n s fk ' each of them be ing e a s i e r to h a n d l e . To quote the

main a p p r o x i m a t i o n t h e o r e m , p r o v e d in sec t ion 3, we assume a lso t h a t

p >1 and t h a t f s a t i s f i e s the c o e r c i v i t y c o n d i t i o n (Co> 0):

(1.s CoI~IPi f(x,s,~)!g(x,s)(l + I~l p)

THEOREM 1.2 - Let f(x,s, ~) b~ a Carath~odory function, quasiconvex

wi th r e s p e c t to ~, and s a t i s f y i n E the g r o w t h condi t ion (1.4) with

p >1. Then there e x i s t s a s e q u e n c e f k ( x , s , ~ ) of Cara th#odory f u n c -

t ions , q u a s i c o n v e x wi th r e s p e c t to ~, and such tha t :

(1.5) Co[~IP<_ fk(x,s,~)~ k(l + [If p) ;

(1.6) fk(x,s,~) : CoI~] p, either fop Isl >_k or I~l> k ;

(1.7) fk <- fk+l ' sup fk = f " k

It is clear that this approximation result is useful to prove

theorem I.i; in fact, in the domain IDu I > k, that is critical of the

integral (i.3), fk reduces to a convex function which is independent

of x and s. One of the difficulties in the proof of theorem 1.2 is

that the definition of quasiconvexity involves an integral inequality

instead that a pointwise inequality such as convexity does. In our

proof we follow a procedure introduced in a similar context by Mar-

cellini and Sbordone [22], and we use a representation formula by

Dacorogna [7], a variational principle by Ekeland [ 14], and a regu-

larity result by Giaquinta and Oiusti [17].

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In section 2 we prove a semicontinuity result for f = f(~), in-

dependent of x and s. The proof is particularly simple and selfcon-

tained. Although this is a special case, it is a crucial step to ob-

tian theorem I.i.

In sections 3 and & we prove theorems 1.2 and i.I respective-

ly.

In section 5 we specialize (1.2). By assuming that f is po~y-

convex In Bali's sense [5] (an example is given by (6.8)), we can

prove a semicontinuity result that, with respect to theorem i.i,

roughly speaking allows us to consider semicontinuity in the weak

topology of H I'p, for p strictly smaller than in theorem i.i. Let us

mention that, in the same context of polyconvex integrals, Acerbi,

Buttazzo and Fusco [i ] proved a semicontinuity theorem in the

strong topology of L , while they have shown a counterexample to

semicontinuity in the strong topology of L p, if p is finite.

In section 6 we give some counterexamples to the semiconti-

nuity theorems I.I and 5.5, when some of the assumptions are not

satisfied.

We thank ].M. Ball and F. Murat for interesting discussions.

2. The c a s e f = f (K)

THOEREM 2.1 - Let f : f({) be a quasiconvex function such that

(2.1) o i f({)!c4 (I + I~l p) ,

for C~ >Oand p > ] . Then the integpal

( 2 . 2 ) I f ( D u ( x ) ) d x

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is sequen t ia l l y lower semicontinuous in the weak topology o f

HI ,p(~ ;~{ N).

We divide the proof of this theorem into 3 steps:

Step I - We assume first that u is affine, i.e., Du K in

nN I] for some KE ~{

Let u h be a sequence in H1,P(a;~{ N) that converges to u in

the weak topology. If u h had the same boundary values as u, then

the semicontinuity result would trivially follow from quasiconvexity

of f. To change the boundary data of Uh, we use a method intro-

duced by De Giorgi [II] , and well known in the context of F-con-

vergence theory (see, i.e. Sbordone [30] and Dal Maso - Modica

[9], theorem 6.1).

Let ~o be a fixed open set compactly contained in n, let

R 1/2 dist (~o, ~), let ~ be a positive integer, and for i :

= 1,2 .... ,~ let us define

i (2.3) l] i = {xE[I: dist (X,ao)<~R } .

"O

Let us choose smooth functions r E Co~(a i) such that

(2.&)

O < ~ < 1 ; r = 1 on ~ ; r : 0 i n 2\ ~. ; - - 1 1 i - 1 1 1

lD~i I <_ (~ +I)/R.

Let us define Vhi = u + ~i(Uh-U). The support of Vhi is con-

tained in ~1; thus by quasiconvexity of f we have

( 2 . 5 )

fa f(Du)dx = f(g)]a[<_ faf(DVhl)dx

f(DVhi)dx + I~] f(DUh)dX = la\aif(Du)dx + fa~ a i-I i-i

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We sum up with respect to i = 1,2,..., v, and we divide by

v. We obtain

1 (2.6) ft~f(Du)dx _< I~ n,f(Du)dx +--v I~ f(DVhi)dx + In f(DUh)dX .

Since DVhi = (l-~i)Du + ~iDUh + (Uh-U)D~i , we have

(2.7)

Ill f(DVhi)dx <- C41~21 + C5{7~ IDulPdx + V

+ /f~lDuhlPd,~ + G~RI)PI lUh-ulPdx}.

Let us go to the limit as h+ + = in (2.6), (2.7). The sequence

Du h is bounded in L p ( a ; N nN), and u h converges s t rongly to u in

Lp(a ,N N). Thus we have

(2.8) Iaf(Du)dx <_ f \ no f(Du)dx + C---!6+ lira inf I f(DUh)dX. v h II

As v -~ + | and ao~, ~ we obtain our result.

Step 2 - f is continuous in the following way:

(2.9) I f ( s ) - f ( n ) [ < C,(1 + I~[ p-1 + In lP-1)]~-n]

(this step is similar to theorem ~.~.i of Morrey [26]). The func-

tion ~: ~ + I~ +, defined by f(~) when only one component (say

E.) of ~ varies, is convex. Thus the derivative r '(~ ) is definite i I

almost everywhere, and we have

(2.10) ~'(~i ) >< (~(~.+h)1 - $(~ ))/hl if h X 0.

For h = +- (I ~I+i) we obtain

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(2.1i)

r + ~(r 1 1

I - - I , ' t ! I f ~ i I~1 + 1

< C (i + [ $ [ p - 1 ) . - - 8

Of course this inequality implies (2.9).

Step 3 - We prove the semicontinuity result for general

u EHI,P(~ ;~ N). Let us consider a partition of a into open cubes

nN i~ (a.n a. = 0 if i/j, a = o ~.) and let us define vectors ~. E~ i i ] i i

by

1 (2.12) ~ = ~ I Du dx .

i

fIN) ~i x E~i'" As the Let us define g E LP(n;R by ~(x) = for

diameter of the partition goes to zero, g converges to Du in

Lp(a;~ nN). Therefore for every E > 0 we can choose the partition

of ~ in such a way that

(2 .13) ~ I IDu - ~ . [ P d x < r �9 12, 1

i 1

Let u be a s e q u e n c e t h a t c o n v e r g e s to u in t he w e a k t o p o l o g y of

H I ' p ( h ; N N ) . F o r e v e r y i , l e t us d e f i n e in a i t h e s e q u e n c e Vh(X) =

= Uh(X) - u ( x ) + < ~ i , x > . As h + + ~ , v h c o n v e r g e s to v ( x ) = < ~ . , x > l

in t h e w e a k t o p o l o g y of H I ' P ( a . ; ~ N ) . T h u s , by s t e p 1, we h a v e 1

(2.1a) lim inf Z f f(DVh)dX > Z f f(g.)dx h a - a i

i I i I

By step 2, and by HSlder's inequality with exponents p/(p-l) and

p , we h a v e

I f f (DUh)dX - Z I f (Dv, ) d x [ < i ~i n - -

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(2.15) < C~ E f a . ( 1 + ]DUhlP-I i i

+ [ D v h l P - t ) [ D u h - ~ i [ d x

<_ Cg ( f n ( l + [ D U h [ + f D V h l ) P d x ) P - I / P ( z i Sa ' I lDu-r e

For the same reason

1/p

(2.16) [ f a f ( D u ) d x - Z f a f ( E 1 ) d x [ < C10e i t

t / p

Our s e m i c o n t i n u i t y r e s u l t is a c o n s e q u e n c e of ( 2 . 1 g ) , ( 2 . 1 5 ) , ( 2 . 1 6 ) .

3. Approximation of quas iconvex functions

In this section we assume, as in (1.g), that f(x,s, ~) is a Ca-

rath@odory function, quasiconvex with respect to ~ , and satisfying

the growth conditions

(3.1) Co[~[ p s f ( x , s ,~ ) s g (x , s ) i + [~IP),

where p >__1, Co> 0, a n d g is a C a r a t h f o d o r y f u n c t i o n . + +

For e v e r y i n t e g e r i , le t r N + N be a c o n t i n u o u s f u n c t i o n

such t h a t r = 1 for 0 < t < i - l , a n d r = 0 for t > i . Let us 1 1 - -

de f ine

(3 .2) g i ( x , s , ~ ) : r + (1 - r ( [ s [ ) ) t C ~

Let A be a subset of a, with zero measure, such that g(x,s) is con-

tinuous with respect to s for every xE a\A. For i,j integers (j>Co),

we define

(3 .3) A., = {xe axA: max {g(x,s) : Is[_< i} < j} . :]

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Aij is a m e a s u r a b l e s e t . We d e f i n e ~ i j ( x ) = t i f x E A i j , a n d ~ i j ( x ) =

=0 o t h e r w i s e , We d e f i n e a l s o

(3 .~) g i j ( x , s , g) = , i j ( x ) g i ( x , s , g ) + (1 - • i j (x ) )Co] K1P

LEMMA 3.1 - For e v e r y i , J , g i j i s a Cara th6odory f u n c t i o n , q u a s i -

c o n v e x wi th r e s p e c t to ~, s a t i s f y i n g :

(3 .5 ) C o l ~ ] P < _ g i j ( x , s , ~ ) < j ( 1 + l~l p) ;

(3.6) g i j ( x , s , K) = ColKI p for ]st >__i;

(3.7) sup gij(x,s,K) = f(x,s, K) , V xE ~\ A, V s, g K- ij

Proo f . g i i is a C a r a t h 6 o d o r y f u n c t i o n , s i nce r is c o n - J

t i n u o u s a n d , i j ( x ) is m e a s u r a b l e . With r e s p e c t to ~' g i a n d g i i ~ a r e

q u a s i c o n v e x f u n c t i o n s : in f a c t t h e y a r e c o n v e x c o m b i n a t i o n of q u a s i -

c o n v e x f u n c t i o n s . I f ]sl > i t h e n gi j = C~ KIP; g i j = ColKl p a l s o if

x ~ A . . ; w h i l e , i f I sl < i a n d x 6 A . we h a v e 1] 1]

(3.8) gij<gi< f <g(x,s)(l+l~l p) _< j(l + l~IP).

Thus ( 3 . 5 ) , (3 .6) a r e p r o v e d . To o b t a i n (3 .7) we o b s e r v e t h a t gii

is nondecreasing with respect to i and j separately, since f >_ gi >

Col ~IP; moreover tim lira gi~ = f" i j J

N nN For every integer m > 1 let us define in a\A • ~ x ~ :

M A R C E L L I N I

(3.9) G.. ( x , s , ~ ) 1]m

g i J ( x ' s ' ~)

[ Co ~I p

for I~ l < m ,

for [ r m ;

(3.10)

gijm(X,S,[) = sup { G(x,s, ~): G is quasiconvex with

respect to [ and G <G.. }. -- lira

LEMMA 3 . 2 - g i j m is q u a s i c o n v e x w i t h r e s p e c t to ~ a n d s a t i s f i e s :

(3.11) C O ~[P < gijm(X,S,~) <_ j(l + I~I p) ;

(3.12) gijm(X,S,~) = Col~I p either for ]~[> m or Isl >_ i.

Proof. Since the supremum of a family of quasiconvex functions

is quasiconvex, in (3.10) we have a maximum and gijm is quasicon-

vex. Since the convex function G = Co[ ~]P is ~eas than or equal to

Gijm, we have Col ~]P<_ gijm' and thus (3.12).

N Fixed x E 2\A and s E~ , we consider the infimum

1 (x,s ~+ Dr CE HIo'P(~;RN)}. (3.13) inf { [-~f Gij m

LEMMA 3.3 - The i n f i m u m in (3.I3) i s a c o n t i n u o u s f u n c t i o n o f (s, O.

Proof. For some fixed x E ~ \A, the function G.. is uniformly N t ]m

continuous for s E ~ and I ~l < m. Thus, for every e>0, there ex-

ists some ~ > 0 such that, if ]s-ti+l~-nl < ~, we have (we decompose

the integral over ~ into two integrals, and we use inequality (2.9)

for the function [ ~I p):

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1 1 G. (x,s r162 - 7 G. (x,t, + D~(y))dy I

Co ( I K+D r ] P._ I n+D~(y) l P )dy i (3.1/4) < e + - ~ , [ a "

< e + C {l r p-I p-I I P-ldy } _ u + [hi + ~T 7~ [ D , ( y ) I 6

Since G.. > Co[ [I p , in the infimum (3.13) we can limit ourself to Ijm -

consider test functions ~ that are uniformly bounded in H~,p(a;~N)1

nN as r varies in a bounded set of IR For all such functions 0o if

Is-tl+IK-nl< min {a, ~ we have

G. ( x , s , K + D r _ 1 G ( x , t , n+D r )dy[< Clue (3.15) I 7,r ljm ~-Tfa ljm

Of cou r se , th i s impl ies t h a t the inf imum in (3.13) is a con t inuous

func t ion of ( s , r

REMARK 3./4 - The previous result does not hold if the integrand

Gij m does not satisfy some properties of structure, such as, for ex-

ample, coercivity with respect to r or continuity on s uniformly with

respect to r (as suggested by corollary 3.12 of [21] ). In fact, if

we consider, as in [21] , G.. = (I+I r I)Isl, then the infimum in tim

(3.13) is equal to G.. if Is I > I, while is equal to 1 if Isl < i. tim

Thus, in this case, the infimum is not continuous (and not even

lower semicontinuous) with respect to s.

LEMMA 3.5 (Dacorogna) - g i j m ( X , S , r i s a Carath6odory func t ion ,

and is equa l to the inf imum in (3 .13) .

Proof. By lemma 3.3 the infimum in (3.13) is a continuous func-

tion of K c ~ nN. It is necessary to use this fact as the first step

in the argument of Dacorogna ( [7], theorem 5; or [8], pag. 87).

Then, like in steps 2,3,g of [7], [8] we obtain that gijm is the in-

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f imum in (3.13). A g a i n by l emma 3 .3 g i j m i s c o n t i n u o u s in (s , ~).

g i j m i s m e a s u r a b l e in x , s i n c e i t i s i n f i m u m of a f a m i l y of m e a s -

u r a b l e f u n c t i o n s .

LEMMA 3 .6 ( E k e l a n d ) - T h e r e e x i s t s a s e q u e n c e u m

m, m i n i m i z e s t h e f u n c t i o n a l s

that, for e v e r y

1 1 (3 .16) ~ +l-a-[ faGijm (x's~<+Dr + - ~ Y a [ D r ( y ) l d y

H 1 N), on ' P ( a ' R a n d s a t i s f i e s 0

1 (3.17) ~i f;iG1]m(X,S,~+Dum(Y))dy < gijm (x's'~) +--'m

P r o o f . T h i s l emma i s a p a r t i c u l a r c a s e of a v a r i a t i o n a l p r i n -

c i p l e given by Ekeland [I~] in the general setting of a complete

metric space V and a lower semicontinuous functional

F : V § ~R u {+ ~} (F ~ + co). Here V = Hl'l(a; ~{N), and F is the inte

gral of G , that is strongly lower semicontinuous in HI'I(a;IRN), ijm

by Fatou's lemma. In (3.17) we use the characterization of gijm giv-

en in lemma 3.5.

The f a c t s t a t e d in ( 3 . 1 6 ) , t h a t u i s a m in imum f u n c t i o n , a i - m

lows us to g e t a M a y e r ' s t y p e r e s u l t [23~, i n t r o d u c e d in t h e c o n t e x t

of m i n i m u m p r o b l e m s fo r v e c t o r v a l u e d f u n c t i o n s by G i a q u i n t a a n d

Giusti [17] (see also [4] for the convex case and [18] for quasi-

minima). Like in lemma 3.2 of Marcellini and Sbordone [22], from

(3.16) we deduce:

LEMMA 3.7 - I f p > 1, t h e r e e x i s t s a n ~ > 0 s u c h t h a t t h e s e q u e n c e

u is bounded in HI'P+~(~}RN ) , m loc

LEMMA 3 .8 - I f p > 1 t h e n o n d e c r e a s i n g s e q u e n c e g i j m c o n v e r g e s

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t o gij a s m + + ~ .

Proof . Let ao be a f ixed open set c o m p a c t l y c o n t a i n e d in a .

The sequence u of the p r e v i o u s lemma is bounded in H I ' P + C ( a o ; N N). m

Thus , if we denote by a the set a = { X e a o : l D u I > m}, we h a v e m m m --

(3.18) : a IDumlPdx -< l amlE/P+ E( : ao lDUmlP+~dx)P /P -~" m

Therefore the left side in (3.18) converges to zero, as m * + ~ . From

(3.17) and (3.5) we obtain

l 1 gi jm +--m > ]-~ f a oGi jm(X ' s ' g+Dum(Y))dY

1 (3.19) >--'~-~:a o\ amgij (x's'$+ Dum(Y))dY

1 >-~-~ : g . (x s , r - --J--: ( I + l D u m ( y ) l P ) d y .

% ~j ' la] a m

Since u is b o u n d e d in H ~ ' P ( a ; NN), it ha s a s u b s e q u e n c e t h a t weak m

l y c o n v e r g e s . We s t i l l denote th i s s u b s e q u e n c e by u , and we denote m

by u e Hlo'P( a ; ~ N) t h e i r weak l imi t . Let m + + = ; we use (3 .18) ,

(3 .19) , the s e m i c o n t i n u i t y r e s u l t of sec t ion 2, and q u a s i c o n v e x i t y

of gij :

--I---7 g+Dum(Y) )dy li%gijm >- limminf I~I ao gij (x's'

(3.20) h]~T:aogij(x,s,~+Du(y))dY

>_ gij(x, s,~) I

l a[ f a x aogij ( x ' s ' ~ + D u ( y ) ) d y

As a o ,~ ~, we obtain our result, since gijm <_ gij "

REMARK 3.9- Several lemmas, from 3.~ to 3.8, are devoted to the

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study of the convergence of gijm as m ~ + | Let us show that this

study is much easier if we know that f is convex with respect to ~:

for every m we can construct a function G(x,s,~), convex with re-

spect to ~, that coincides with gij for I~I <_ m and grows linearly

at = (the supremum of all hyperplanes supporting gij where I~ I < m).

Since G grows linearly, there exists an m' > m such that G<_CoI~I p

for I ~I _ > m'. By the same definition (3.10), G -gijm < -< gij' and thus

gijm' = gij for I~I <_ m. Note that also in this simple argument for

the convex case we need p >i.

Finally we obtain the approximation result stated in the in-

troduction:

fine

Proof of theorem 1.2 - For every integer k (>_ 2 + C o ) we de-

(3.21) fk(x,s,r = max {gijm(X,S,~) : i + j + m <_k } .

(1.5) is consequence of (3.11), while (1.6) follows from (3.12).

Finally the supremum of fk is f, by lemma 3.8 and formula (3.7).

4. Semicontinuity in the quasiconvex case

In this section we will prove theorem I.i. It will be conse-

quence of the approximation theorem 1.2., by proving a semiconti-

nutty result for the functions fk as in theorem 1.2. In the following

lemmas we assume k is fixed and fk satisfies (1.5). (1.6).

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LEMMA 4.1 (Scorza-Dragoni) For e v e r y p o s i t i v e r t h e r e e x i s t s a

compact set C c ~, with l a\C[<r such that fk(x,s,{)

N nN in C x~. x ~

is continuous

Proof. See i.e. lemma 1 of pag. 37 of [i0], or [15], chap.

Vlll, section 1.3.

LEMMA 4.2- There exists a continuous bounded function w:~ +§ ~+,

with w(0)=0 , such that, for x,yE C, s,t E~9 N, ~ E ~9 nN we have

(4.1) I f k ( x , s , C) - f k (Y , t ,C) l <_w( l • § I s - t l ) �9

Proof. For i~[<_ k, by (1.5), we have

(4.2) I fk (x , s , 0 - fk (Y , t , ~)[ ! 2 k ( 1 + k p)

Inequality (4.2) holds also if [ ~I > k since the left side is zero.

The function fk is continuous in the compact set C • {Is[ <__ k+l} •

• {[ ~[ < k+l }. Thus (4.1) holds on this set with w equal to the

oscillation of fk" By (4.2) the function w is bounded, and we can

assume that w(r) = 2k(l+k p) for r >__I. By 1.6), formula (4.1) holds

also if either I~[ >__ k or [sl and [t[ >__ k. It remains to consider the

case [s[ < k, [tl > k+l and [~[ < k. In this case w = 2k(l+kP), and

thus (4.1) follows from (4.2).

LEMMA /4.3 - The i n t e g r a l

(~.3) s fk(x,u(x) Du(x))dx

I

is sequentiaTly lower semicontinuous in the weak topology of H~'P(a;~N).

Hl,P( N). Proof. Let u E a ;~ Let us consider a partition of

n ~. = 0. u ~ ~. Like in Morrey n into open cubes ~i' with ~i ] i i =

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[ 2 5 ] w e d e f i n e t h e v e c t o r v a l u e d f u n c t i o n s ( c o n s t a n t i n e a c h a . ) : I

1 1 : x d x �9 u - I u ( x ) dx . ( 4 . ~ ) x i ~T-s ' 1 [ a . I a �9

I 1 1 I

By t h e d o m i n a t e d c o n v e r g e n c e t h e o r e m , f o r e v e r y e > O, we c a n

choose the partition so that

w ( l • 2 1 5 + l u ( • < ~. (4 .5 ) z a 1

Let u h be a s e q u e n c e of H l , P ( a ; ~ 9 N) t h a t c o n v e r g e s to u in t h e

w e a k t o p o l o g y . We h a v e

/ fk(X,Uh,I)Uh)dX = f \C{fk(X,Uh,DU h) - fk(xi,ui,DUh)} dx

(&.6] + YC { f k ( X ' U h ' D U h ) - fk(X'u'DUh )} dx

+ IC { fk(x'u'Duh ) - fk(xi'ui'DUh )]dx + /~fk(xi'ui 'Duh)dx

We use (&.2), (5.1) and (4.5);

(4.7)

f fk(X,Uh,PUh)dx >- 2k(l+kP)I ~\ CI

-ICw(lUh-U/)dx.. - a + Z I~ fk(xi,ui,DUh)dX i l

As h -* + =, by the semicontinuity theorem 2.1, we have

(4.8)

zero.

lira inf I fk(X,Uh,DUh)dX> - C~3E + l~fk(xi,ui ~)u)dx h ~

We obtain the proof as the sides of the cubes a. and ~ go to

Proof of theorem i.i - Let us assume first that p >i. Similarly

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to S e r r i n [ 3 1 ] , f o r e > O, l e t us d e f i n e

(~ .9 ) g ( x , s , 0 = f(x s r + % l s l t , , + C , ( x ) + e l g ] P + C . g

S i n c e p > r , we c a n c h o o s e the c o n s t a n t C to o b t a i n g ( x , s , ~ ) > g E - -

>_ ~/2 I g[ p . Let f e k be the s e q u e n c e of q u a s i c o n v e x f u n c t i o n s t h a t

c o n v e r g e s , a s k + + ~, to g , a c c o r d i n g to t h e o r e m 1 .2 . I f u h w e a k - 1

l y c o n v e r g e s to u in H ' P ( a ; N N), b y l emma ~.3 we h a v e

(~ .10)

l i m h i n f : a g e ( x ' U h ' D U h ) d X - > limk l i m h i n f : a f a k ( X , U h , D U h ) d x

l ira : faf k ( X , u , D u ) dx = : a g ( x , u , D u ) d x .

k

Let C~4 be a n u p p e r b o u n d fo r the H I ' p norm of u h . S i n c e u h

c o n v e r g e s to u in t he s t r o n g t o p o l o g y of Lt(~;1R N) ( h e r e we use t he

a s s u m p t i o n t h a t aa is smooth i f C 2 / 0 ) , we o b t a i n

l im i n f : a f ( x , U h , D U h) dx > l im i n f : a g (X ,Uh ,DUh)dX h - h

t ~ cp (&.11) - l im : a { C 2 [ U h l + C~ (x) + C } dx - 7 h

~ c~ : a f ( x , u , D u ) d x + - ~ - : a I D u t P d x - 7 "

We complete the proof of the case p >i as ~ + 0. If p = I, the proof

is much simpler since, if UhConverges to u in the weak topology of

Hl,l(2;~ N), then the integrals of IDUhl are equiabsolutely contin-

uous. We do not give the details; we can use the approximation lem-

ma 3.I and then the argument by Fusco [16 ], or the argument of

section 2 of [22].

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5. The polyconvex case

In this section we consider a particular case of quasiconvex

functions. Following Ball [5], we say that a function f(x,<) is po-

7yconvex with respect to < if there exists a function g(x,<), convex

m with respect to ne R , such that

(5.1) f(x, 0 = g(x,~,det~, det2~ .... ) ,

where det ~ are subdeterminants (or adjoints) of the n x N matrix i

~. If ~ = Du, then each determinant is a divergence. For example,

for n = N = 2, if u = (u~,u~) aC2(a:~2), we have

(5.2) det Du = u l u 2 u I u 2 = (ulu 2 ) - (ulu 2 ) X I X 2 X 2 X I X 2 X i X I X 2

Using (5.2) (and in general (5.~)), we can verify by Jensen's ine-

quality that every polyconvex function is quasiconvex. By multi- co

plying (5.2) by a test function ~ e C o(~) and by integrating by

parts we have

(5.3) f det Du ~dx = - Ia{ulu2 ~ - u ~ u 2 r } dx x 2 x~ x~ x~

By continuity (5.3) holds for u E HI'2(~;~ 2 ). If u EHI'P(~;]R 2) for

, L 2p/2-p Ux2 p < 2 then u E loc (~ ;R 2) and thus the product u I 2 is sum-

mable if I/p + (2-p)/2p <_i, i.e., p >_4/3. Thus, if uEHI'd/3(a;~ 2 )

we can define by (5.$) the determinant of Du as a distribution.

Moreover for the same reasons the map u § det Du is continuous in

the following sense: if u h converges to u in ~oP(~;~2 ) for p> 4/3

and if (5.3) holds for u h and u, then det Du h converges to det Du

strongly con- in the sense of distributions. In fact in this case u h

verges to u in L ~ (~) and we can go to the limit as h § + ~o in (5.$).

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For general n = N >_2 we can still write d e t Du as a diver-

gence (Morrey [26 ], pp. 122-123)

~(Ul , .. U n) n ~ n) �9 , ~ (u 2 ,... ,u (5.~) det Du= ~(xl ' ,x )= - z (-i) m- (uX~(x ..,x _1, x ..,x )

n m=l ~ x ~ ' ~-W n

This formula holds if uG Hl'n(~ ;JR n). Since if uE HI'P(n-I_< p< n)

then uX e L np/(n-p) loc and the ]acobian of order n-i belongs to

L p/(n-l), the right side of (5.&) is well defined in the sense of

distributions if (n-l)/p+(n-p)/np _< i, i.e., p >_ n 2/(n+l). Thus we

have proved, as in Ball [5] and Ball, Currie and Olver [6 ], the

following result :

- L e t n = N>_2and U@~ol(Q;~n).l_n If LEMMA 5.1 ([5],[6])

p>_n2/(n+l) then det Du is defined by (5.&) as a distribution;

L 1 - f u n c t i o n a n d f o r m u l a w h i l e i f p > n , d e t Du i s d e f i n e d as a loc

(5./~) h o l d s . M o r e o v e r , i f u h c o n v e r g e s to u in the w e a k t o p o l o g y

of Hll'oc P for p >n'/(n+l) , then det Du h converges to det Du in

the sense of distributions.

To get a semicontinuity result for polyconvex functions, we con-

sider a function g(x, n) defined for xE~ C ~ n and nER m with val-

ues in [0,+~] (see in next section the reason to assume g continuous

in x and independent of s) satisfying:

(5.5) The set {(X,n) : g(x,n) < + ~ } is open {and not empty)

m in ~ ~ ~{ and g is continuous on this set.

(5.6) g(x, n) is convex and lower semicontinuous with respect

m to n E ~{

REMARK 5.2 - Of course, the semicontinuity assumption in (5.6) is

a nontrivial condition only at the boundary of the domain where g

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MARCELLINI

is finite. We assume (5.5) to simplify the next lemma, but we could

consider also other eases. On assuming (5.5), (5.6) we have in mind

the situation described by Ball [5 ], of interest in nonlinear elas-

ticity (see also the paper by Antman [3] ), where are considered

functions f(x,Du) = g(x,det Du) that are finite if and only if

det Du > 0, and go to + ~ if det Du § 0.

Let us begin with two approximation lemmas.

LEMMA 5-3 (De Giorgi) - Thepe exists a nondecr'easing sequence of

peal nonne~ative functions gk that conver'ge, as k § ~ , to g. For

m every k,gk(X, n) is uniformly continuous in ~ • ~ ,it @rOwS lin-

early with respect to n, it is convex with respect to q and it is

equal to zero if dist (x,as

Proof. Let a i = {xE 9 : dist (x,~) > 1/i}, and let ~i ~ C[(a)l

be e q u a l to one on a i_ l , a n d Oi ~ 0 . Fo r e v e r y x e ~ , t h e f u n c t i o n m

g is lower semicontinuous on R ; thus it is the supremum of a se-

quence of affine functions ( a . ( x ) , n) + b . ( x ) . L ike in p a g . 31 of De ] ]

G i o r g t [ 1 0 ] ( the a r g u m e n t of [ 1 0 ] c a n be a p p l i e d , s i n c e g i s f i n i t e

in a n e i g h b o u r of e a c h s u p p o r t i n g p o i n t ) , we c a n c h o o s e a . ( x ) a n d ]

b . ( x ) to be c o n t i n u o u s in a. Let us d e f i n e a0 , b0 -- O, a n d ]

(5.7) gk(X, n) : max {~.(x)[ (a (x), n) + b (x)] : i+j< k} i ] ] -

The sequence gk has all the required properties.

r

LEMMA 5.4 - There exists a sequence hk(X,n) of C -functions sat-

isfyin~ all the pvopepties stated in the previous lemma (except the

fact that hk> - -i).

co

Proof. Let be a mollifier, i.e. ~ECo(~{ n • ~9 m), I c~ dxdy : 1,

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MARCELLINI

-(n+m) c~ >0._ Let us define h k r : hk.~e, where c~e (x,n):e a(x/e,n/e).

I f s i s s u f f i c i e n t l y s m a l l , h k r i s a n o n n e g a t i v e C - f u n c t i o n , a n d

i s z e r o i f d i s t ( x , ~ a ) _ < 1 / k + l . By t h e u n i f o r m c o n t i n u i t y of g k ' a s m

r +0 h ke converges to gk uniformly in a • ~ . Thus we can choose

r = e ( k ) s u c h t h a t

1 (5.8) l hk,e(k) - gkI<2(k+l)2

F o r k > 2 l e t u s d e f i n e h k ( X , n) = h k _ l , e ( k _ l ) ( x , n ) - ( k - l ) -1 . The

s e q u e n c e h k s a t i s f i e s a l l t h e s t a t e d p r o p e r t i e s . F o r e x a m p l e , l e t u s

v e r i f y t h a t h k i s i n c r e a s i n g w i t h r e s p e c t to k :

1 1 1 1 h k < < _ . g k - 1 +- '2-~ k - i - - g k + "2k 2 k - 1

1 1 1 (5.9) < h

k , E (k ) + ~ +- ' i -k -r - k - I

1 1 1 < +--~ + < .

hk+l ~ k-I hk+l

Now we prove a semicontinuity result, that generalizes an a-

nalogous result of Reshetnyak [28 ].

THEOREM 5.5 - Let g: a • ~{n+ [0,+,] be a function satisfyin@ (5.5),

(5.6) . Let v h and v be functions of L 11oc(a;~m), and assume that

v h converges to v in the sense of distributions, i.e. :

(5.10) lim f

h

a (v h' ~)dx : f (v, r , ~ ~EC~(~;~9 m)

Then w e have

( 5 . 1 1 ) limhinf /a g(X,Vh(X))dx _> Iag(x,v(x)) dx

Proof. By using our approximation lemma 5.g in the usual way

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MARCELLINI

like in (&.lO), it is enough to prove the theorem assuming that g

is C "(n • R m), and that there exists an open set ~o compactly con-

tained in ~ such that g is equal to zero for x (~ ~0. Let ~ be a c

mollifier and let v = v , %. By the convexity of g, similarly to

Serrin [31 ], we have

(5.12) g(x,v h) >_g(x,v ) +(D g(x,v ), Vh-V ~)

Since D g(x,v (x))6 Co~(a;~{m), by (5.10) we have n e

(5.13) lim inf I g(x,Vh)dX >7 g(x,v )dx+/ (D g(x,v ),v-v )dx h n - ~ ~ n ~ e

We obtain the result for c -~ O. In fact, in the first term of the

r i g h t s i d e we c a n use F a t o u ' s l e m m a , w h i l e in t h e s e c o n d t e r m we

c a n u s e t h e f a c t t h a t D g is b o u n d e d in a x R m i n d e p e n d e n t l y of n

By lemma 5.1 and theorem 5.5 we obtain two semicontinuity re-

sults for integrals of the type:

(5.14) F(u) : I n f(x,Du)dx = I g(x,Du,det Du, det 2 Du .... )dx

Here f(x, ~) is a polyconvex function like in (5.1), and g(x,n) sat-

isfies (5.5), (5.6).

COROLLARY 5 .6 - Let u h and u he functions of H1~7 ( R ; q R l r ~ N) , fo r P >

> rain {n2/(n+l);Nn/(n+l)}. Assume that the subdeterminants of the

Jacobians Du h and Du ape defined as L ~ -functions and that for- loc

mula (5.&) holds for u h and u, If u h converges to u in the weak

topology of H~ocF(~;~N),1_ then lira inf F(u h) >F(u). h

COROLLARY 5.7 - Let u h and u be function of Hl'r(loc ~; ~ N) . for r _>

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MARCELLINI

> m i n { n , N } . I f u h converges to u in the weak topology of H l o c ( a ; ~ N ) 1 . o

fo r p > m i n { n 2 / ( n + l ) ; N n / ( n + l ) } then l i ra i n f F ( u h) > F ( u ) . h

6. Some examples and remarks

Here we d i s c u s s t h e n e c e s s i t y of some a s s u m p t i o n s of t h e s e m i -

c o n t i n u i t y t h e o r e m s 1 .1 a n d 5 . 5 . Le t u s b e g i n w i t h t h e o r e m 5 . 5 a n d

l e t u s s h o w t h a t t h e r e s u l t d o e s n o t h o l d i f g ( x , n) i s o n l y m e a s u r -

a b l e w i t h r e s p e c t to x , o r i f g = g ( x , s , n ) ( w i t h t h e u s u a l m e a n i n g

of s ) . Of c o u r s e , to e x h i b i t c o u n t e r e x a m p l e s , we m u s t c o n s i d e r n o n

c o e r c i v e c a s e s : i n f a c t , i f g ( x , n ) > c o s t In l p f o r some p > 1 , t h e s e -

m i c o n t i n u i t y t h e o r e m 5 . 5 r e d u c e s to t h e u s u a l s e m i c o n t i n u i t y t h e o r e m

i n t h e w e a k t o p o l o g y of L p .

EXAMPLE 6 .1 - Let n : m = 1 a n d l e t g ( x , n) = a ( x ) n 2, w i t h a ( x )

n o n n e g a t i v e , b o u n d e d a n d m e a s u r a b l e i n ( 0 , 1 ) . I t h a s b e e n p r o v e d

i n t h e o r e m 5 of [19] t h a t f o r e v e r y p e [ 1 , + - ] a n d fo r e v e r y ueH 1 ' 2

t h e r e e x i s t s in H 1 ' 2 a s e q u e n c e u h t h a t c o n v e r g e s to u in L p a n d

satisfies

(6.1)

where

1 1

lira I 0 a(x) (Uh)2dx = 70 b(x) (u')2dx ,

h

(6.2) b(x) = lim 2e[f x+r (y)dy]-1 X--E

E+ 0 +

If we consider a function a(x) { 0 a.e. that is not locally sum-

mable a.e. in (0,i), then b(x) is zero a.e. Let us define v h = u h

and v = u'. Then, for every ~>~ C7, we have

( 6 . 3 ) lira Io ~ v h ,dx : - lim I~ u h r = -Io ~ u~'dx : Io ~ v~dx. h h

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MARCELLINI

M o r e o v e r v h a n d v a r e i n L 1, b u t t h e y do no t s a t i s f y ( 5 . 1 1 ) , s i n c e ,

i f v i s no t i d e n t i c a l l y z e r o ,

(6.&) lim I o a(x) v~(x) dx : O< fo a(x) v~(x)dx h

EXAMPLE 6 .2 - We c a n a d a p t t h e c o u n t e r e x a m p l e of E i s e n [ 1 2 ] to

o u r s i t u a t i o n . In f a c t , E i s e n s h o w e d t h a t t h e r e e x i s t s a s e q u e n c e

of L i p s c h i t z c o n t i n u o u s f u n c t i o n s u h =- (U~h,U~) d e f i n e d in ( 0 , 1 ) ,

s u c h t h a t t h e p r o d u c t u ~ ( u ~ ) ' = 0 a . e . , a n d u h c o n v e r g e s in L ' to

t h e f u n c t i o n u ( x ) - ( 1 , x ) . Let u s d e f i n e v h = ( u ~ ) ' , v = ( u ~ ) ' = 1,

w h = u~ , w = u ~ = 1. L ike in ( 6 . 3 ) , v h c o n v e r g e s to v in t h e s e n s e

of d i s t r i b u t i o n s ; v h a n d v a r e i n L ~ , b u t

1 1

(6.5) I~ (WhVh)2dx = 0 , I o (wv) 2dx = i

T h i s m e a n s t h a t i n g e n e r a l we c a n n o t e x t e n d t h e o r e m 5 .5 to i n t e -

g r a l s of t h e t y p e I g ( w ( x ) , v ( x ) ) d x , w h e r e g ( s , n) i s c o n t i n u o u s in

( s , n) , a n d c o n v e x w i t h r e s p e c t to n , a n d t h e t o p o l o g y c o n s i d e r e d

i s t h e p r o d u c t of t h e L ~ no rm t o p o l o g y fo r w, a n d t h e t o p o l o g y of

d i s t r i b u t i o n s fo r v .

EXAMPLE 6 ,3 In t h e o r e m 1.1 t h e a s s u m p t i o n t < n p / ( n - p ) i f p < n

i s n e c e s s a r y . We h a v e a c o u n t e r e x a m p t e fo r f ( x , s , ~) = I < I p -

- c o s t l s l n p / ( n - p ) , b y c h o o s i n g a s e q u e n c e u h t h a t w e a k l y c o n v e r g e s

in H I ' p , b u t d o e s no t c o n v e r g e in t h e no rm t o p o l o g y of L n p / ( n - p ) .

EXAMPLE 6 .4 - I f n a n d N a r e g r e a t e r t h a n o r e q u a l to 2, t h e a s -

s u m p t i o n r < p in t h e o r e m 1.1 i s n e c e s s a r y . In f a c t t h e r e is a c o u n -

t e r e x a m p l e b y M u r a t a n d T a r t a r ( s ee t h e c o u n t e r e x a m p l e in s e c t i o n

4.1 of [27]), for n = N = p = r = 2, where it is shown that the in-

tegral

24

MARCELLINI

(6 .6 ) f a ( x ) d e t Du dx ,

is not continuous in the weak topology of HI,2(a;IR 2 ), even if a

is a nonzero constant. It is a consequence of theorem 1.i, but it

is also well known (see [29 ], [5], [6]; and for more general func-

tionals [16], [22]), that, if ac L'(a), the integral in (6.6) is

weakly sequentially continuous in the weak topology of H l'2+e (a;}{2),

for every positive e.

EXAMPLE 6.5 - To discuss the necessity of the upper bound in (i.i)

let us summarize our results in the special case n = N >_ 2 for the

integral

(6 .7 ) 7 a g ( x , u ( x ) ) l d e t D u ( x ) l a dx

We distinguish two cases: ~ _> 1 or (~ < 0; in the secondcase we de-

fine I nl ~ = + ~ if n < O. If g is a nonnegative Carath6odory func-

tion and c~ >_ I, then the integral (6.7) is sequentially lower semi-

continuous in the weak topology of Hl,n(~;~ n). This follows from

theorem I.i in the general case (if a > 1 we can approximate Inl ~

with a nondecreasing sequence of convex functions on ~{ , each of

them growing linearly at -), and from corollary 5.7, if g is inde-

pendent of s and continuous. If g = g(x) is a nonnegative continu-

ous function, and either ~ > 1 or ~ <0, then from corollary 5.7 it

follows also that, if p> n2/(n+1):

(6.8) If u h and u are smooth {say Uh'UE Hl,n(~;~9 n)) and u h

weakly converges to u in Hl'P(a;~ n) , then

lira inf f~ g(x)l det DUh la dx _> Jag(x)Idet DuJ a dx.

h

Let us mention explicitely that we have not proved that (6.8)

25

MARCELLINI

is true if l< p <_n2/(n+l). Let us also mention that Acerbi, Buttaz-

zo and Fusco [i] proved that (6.8) is not true if we replace the

weak convergence in Hl,P(a ;R n) with the strong convergence in

LP(a;~n), whatever pe [I,+ ~. ) may be.

REMARK 6.6 - In (6.8) we distinguish between the space where the

functional is well defined, and the space where the sequence u h

weakly converges. This is a natural point of view and it is not

new. In this context of polyconvex functions we refer to theorem

9.2.1. by Morrey [26 ], and to [I]. We refer also to the well known

semicontinuity theorems by Serrin [31] (see also [26], section 4.1),

where the considered functions u h are required to be in the space

H I'I, but the convergence is in the space L I We refer also to the

theory of De Giorgi [II ], related to this subject.

REFERENCES

[I] ACERBI E., BUTTAZZO G., FUSCO N., Semicontinuity and relaxation

for integral depending on vector-valued functions, J. Math. Pures

Appl., 62 (1983), 371-387

[2] ACERBI E., FUSCO N., Semicontinuity problems in the calculus of

variations, Arch. Rat. Mech. Anal., to appear

[3] ANTMAN S.S., The influence of elasticity on analysis: modern de-

velopments, Bull. Amer. Math. Soc., 9 (1983), 267-291

[4] ATTOUCH H., SBORDONE C., Asymptotic limits for perturbed function-

als of calculus of variations, Ricerche Mat., 29 (1980), 85-124

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Paolo Marcellini

Dipartimento di Matematica

II UniversitA di Roma

Via Orazio Raimondo

00173 ROMA, Italy

(Received February 28, 1984)

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