Approximation of quasiconvex functions, and lower semicontinuity of ...

m a n u s c r i p t a math.

manuscripta mathematica

51, I - 28 (1985)

9 Springer-Verlag 1985

APPROXIMATION LOWER

OF QUASICONVEX

SEMICONTINUITY

Paolo

FUNCTIONS,

OF M U L T I P L E

AND

INTEGRALS

Marcellini

We study semicontinuity of multiple integrals ~ f ~ , u , D u ) d x ~ where the vector-valued function u is defined for x E ~ C N with values in ~ . The function f(x,s,~) is assumed to be Carath@odory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower 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 values 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 ed

open

us set

consider ~ of

Carath~odory continuous

(1.1)

Here

growth p>l;

with

2

r

>0;

1 a n d 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~lp)

THEOREM

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

with

respect

to

~,

p >1.

Then

tions,

quasiconvex

and

s a t i s f y i n E the g r o w t h

there exists

a sequence

w i t h r e s p e c t to

fk(x,s,~) ~, a n d s u c h

(1.5)

Co[~IP k, that is critical of the

to a convex

of quasiconvexity

Sbordone [7],

with

(1.4)

sup fk = f " k

that

that the definition instead

'

in fact,

s.

either fop Isl >_k

: CoI~] p,

condition

[17].

Ekeland

formula

[ 14], and

by

a regu-

MARCELLINI

In section 2 w e dependent tained.

of x

and

Although

tian theorem

prove

a

s. The

semicontinuity

result for f = f(~), in-

proof is particularly simple a n d

this is a special case,

selfcon-

it is a crucial step to ob-

I.i.

In sections 3 a n d

& we

prove

theorems

1.2 a n d

i.I respective-

ly. In section 5 we convex In Bali's prove

a

specialize (1.2).

sense

[5]

semicontinuity

roughly

speaking

allows

topology

of H I'p,

for p

mention

that,

Buttazzo

and

strong

in

the

Fusco

topology

of L

(an

result us

By assuming

e x a m p le that,

strictly

respect

context

[i ]

proved they

a

to

in

shown

i.i,

the w e a k

i.i. Let us

integrals,

semicontinuity have

theorem

in theorem

of polyconvex

po~y-

(6.8)), we can

semicontinuity

smaller than

same

, while

is given by

with

to consider

that f is

theorem

Acerbi, in

a counterexample

the to

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

section

theorems

6

we

I.I

and

give 5.5,

some when

counterexamples some

of the

to the

semiconti-

assumptions

are

satisfied. We t h a n k ].M.

2.

The case

Ball a n d

2.1 - Let

(2.1)

o i f({)!c4

(2.2)

for interesting discussions.

f = f(K)

THOEREM

for

F. Murat

C~ >Oand p > ] .

I f(Du(x))

f : f({) be a quasiconvex function such that

(I + I~l p)

,

Then the integpal

dx

not

MARCELLINI

is

sequentially

lower

semicontinuous

in

the

weak

topology

of

HI ,p(~ ;~{ N).

We divide the proof of this theorem

into 3 steps:

Step

is affine,

I - We assume nN I] for some KE ~{ Let the w e a k the

uh

be

a

sequence

topology.

result

of f.

To

change

duced

by

De Giorgi theory

[9], theorem

R

Let

~o

1/2

dist

in

If u h had

semicontinuity

vergence

first that

the

H1,P(a;~{ N) the same

would

data

[II] , and i.e.

follow

well k n o w n

Du

K

that converges

of Uh,

Sbordone

i.e.,

boundary

trivially

boundary

(see,

u

values from

we

use

and

to u in

as u, then

quasiconvexity a method

in the context

[30]

Dal

in

intro-

of F-con-

Maso

-

Modica

6.1). be

a

fixed

(~o,

~),

open let

~

set be

compactly a

positive

contained integer,

in

and

n,

let

for i :

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

(2.3)

i dist ( X , a o ) < ~ R"O

l]i = {xE[I:

} .

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

O< ~ --

< (~(~.+h)1

For h =

+- (I ~I+i) we obtain

have

- $(~ ))/hl

if h

X

0.

MARCELLINI

r

+

1

I --I,'t ! If~i

I~1

~(r

1

+ 1

(2.1i)

< C (i + --

[$[p-1).

8

Of course this inequality implies

Step u

EHI,P(~

i~ ( a . n i i by

(2.12)

~

diameter Lp(a;~

We

;~ N).

i

prove

Let

1

= ~

define

of

the

be

HI'p(h;NN).

in t h e

let us

fIN) b y

~(x)

for

general

a into open

define vectors

cubes nN ~. E ~ i

goes

to

= ~i

zero,

E > 0 we

g

for x E~i'"

converges

can

choose

to

As

the

Du

in

the partition

that

IDu -

~.[P

dx
Z f f(g.)dx a i i I

inequality

we h a v e

If

f(DUh)dX

-

Z I f(Dv, )dx[ i ~i n

< - -

with

exponents

p/(p-l)

and

MARCELLINI

(2.15)

< C~ E

fa.(1

i

i

+ ]DUhlP-I

+ [DvhlP-t)[Du h -

Co[ [I,p in the infimum (3.13) we can limit ourself to Ijm consider test functions ~ that are uniformly b o u n d e d in H~,p(a;~N)1 nN as r varies in a b o u n d e d set of IR For all such functions 0o if

Is-tl+IK-nl< (3.15)

min

{a,

~

we have

7,r G.ljm ( x , s , K + D r

I

Of c o u r s e ,

this

implies

that

_ ~-Tfa 1

the

infimum

Gljm ( x , t , n+D r

in

(3.13)

not

hold

)dy[< Clue

is a c o n t i n u o u s

f u n c t i o n of ( s , r

REMARK Gij m

3./4 -

does

ample,

previous

not satisfy

some

result

to

consider,

r (as suggested

by

of structure,

if the such

integrand

as, for ex-

r or continuity on s uniformly corollary

3.12 of

with

[21] ). In fact,

if

[21] , G.. = (I+I r I)Isl, then the infimum in tim is equal to G . . if Is I > I, while is equal to 1 if Isl < i. tim in this case, the infimum is not continuous (and not even

(3.13) Thus,

as

does

properties

coercivity with respect to

respect we

The

in

lower semicontinuous)

LEMMA 3.5 ( D a c o r o g n a )

with respect to s.

- gijm(X,S,

r

is

a

Carath6odory

function,

and is e q u a l to the i n f i m u m in (3.13).

Proof. tion of in

the

Then,

By l e m m a

K c ~ nN. argument

3.3 the infimum

It is necessary of Dacorogna

like in steps

2,3,g of

in (3.13) is a continuous

to use

( [7],

11

this fact as the first step

theorem

[7], [8] we

func-

5; or [8],

obtain

pag.

87).

that gijm is the in-

MARCELLINI

fimum gijm

(3.13).

in is

urable

Again

measurable

in

by x,

lemma since

3.3

gijm

is

it

is

infimum

exists

a

sequence

continuous of

a

family

(Ekeland)-

There

u

minimizes

the

1

~ +l-a-[ faGijm (x's~ 1

the

nondecreasing

12

sequence

gijm

converges

MARCELLINI

to

gij

as

m

Proof.

+ ~

Let

The s e q u e n c e Thus,

+

.

ao be

a fixed

open

set c o m p a c t l y

contained

in a .

u

of the p r e v i o u s lemma is b o u n d e d in H I ' P + C ( a o ; N N). m if we denote b y a the set a = { X e a o : l D u I > m}, we h a v e m m m -:a

(3.18)

IDumlPdx

-< l a m l E / P + E ( : a o l D U m l P + ~ d x ) 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

gijm +--m > ]-~ f a o G i j m ( X ' s ' g + D u m ( Y ) ) d Y 1 >--'~-~:a o\ amgij (x's'$+ Dum(Y))dY

(3.19) 1

>-~-~ :

Since u

m

ue

(3.19),

g .(x s , r

is b o u n d e d

ly converges.

by

%

- --J--:

la]

'

in H ~ ' P ( a ; NN),

a

(I+lDum(y)lP)dy

.

m

it h a s a s u b s e q u e n c e

t h a t weak

We s t i l l denote this s u b s e q u e n c e

Hlo'P( a ; ~ the

~j

N) t h e i r

semicontinuity

weak

by u , a n d we denote m Let m + + = ; we use (3.18),

limit.

result

of

section

2,

and

quasiconvexity

of gij :

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

(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 O, we c a n

so that

+ lu(•

sequence

of

< ~.

1

Hl,P(a;~9

N)

that

converges

to

u

in

the

We h a v e

/ fk(X,Uh,I)Uh)dX

(&.6]

u

(constant

1

dominated

topology.

9 '

functions

1

choose the partition

(4.5)

valued

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

+ YC { f k ( X ' U h ' D U h )

dx

- fk(X'u'DUh )} dx

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

We use

(&.2),

(5.1) and

f fk(X,Uh,PUh)dx

(4.5);

>-

2k(l+kP)I ~\ CI

(4.7) -ICw(lUh-U/)dx..

- a + Z I~ fk(xi,ui,DUh)dX i l

As h -* + =, by the semicontinuity

(4.8)

theorem

lira inf I f k ( X , U h , D U h ) d X > h ~ We

obtain

the proof

as the

2.1, we have

- C~3E + l~fk(xi,ui ~)u)dx

sides

of the cubes

a. and

~ go to

zero.

Proof of theorem

i.i - Let us assume

16

first that p >i. Similarly

MARCELLINI

to S e r r i n

(~.9)

Since

[31],

for

g (x,s,

0

p > r,

>_ ~/2

I g[ p .

e > O, l e t u s d e f i n e

= f(x , s , r

we

can

choose

Let

fek

be

+ %lsl t

the

the

+

C,(x)

elg]P

+

Cg

to o b t a i n

of q u a s i c o n v e x

functions

a s k + + ~, to g , a c c o r d i n g to t h e o r e m 1 . 2 . 1 l y c o n v e r g e s to u i n H ' P ( a ; N N), b y l e m m a ~.3 we h a v e

:age(x'Uh'DUh)dX-

.

gE (x,s,~)

converges,

limhinf

C g

constant

sequence

+

>

--

that

If u h w e a k -

> limk l i m h i n f : a f a k ( X , U h , D U h

)dx

(~.10) lira :faf k ( X , u , D u ) k Let C~4 be converges

an

to u i n

assumption

that

the

bound

strong

for

(x,u,Du)

the

since,

if UhConverges

then

section 2 of

[22].

the

argument

of IDUhl

by

17

we u s e t h e

(X,Uh,DUh)dX

-

~ cp 7

- 7 "~ c~

to u in the w e a k

do not give the details; we can and

Since u h

as ~ + 0. If p = I, the proof

uous.

3.I

the integrals

of u h.

we o b t a i n

+-~-:aIDutPdx

N),

ma

norm

.

of Lt(~;1R N) ( h e r e

Hl,l(2;~ We

then

HI ' p

+ C } dx

the proof of the case p >i

simpler

dx

d x > lim i n f : a g h

: a { C 2 [ U h l t + C~ (x)

:a f(x,u,Du)dx

is m u c h

:ag

topology

: a f ( x , U h , D U h)

- lim h

We complete

=

aa is smooth i f C 2 / 0 ) ,

lim i n f h (&.11)

upper

dx

Fusco

are

topology of

equiabsolutely

contin-

use the approximation [16 ], or the

lem-

argument

of

MARCELLINI

5. T h e

polyconvex

In

this

functions.

section

with

respect

(5.1)

to

where

det

~.

~ = Du,

i

ne R

~ are

quality

then

each =

every

that

case

of q u a s i c o n v e x

a function

a function

f(x,

uh

in 4/3

to det D u

strongly

go to the limit as h § + ~o in

18

2 )

distribution.

Du h converges

this

2) for

if u E H I ' d / 3 ( a ; ~

to u in ~ o P ( ~ ; ~ 2

then

In

u I U x22

the p r o d u c t

p >_4/3.

converges

of distributions.

to u in L ~ (~)

~

determinant

reasons

for

x2

for u E H I ' 2 ( ~ ; ~

+ (2-p)/2p _2 we

[26 ], pp.

still write d e t

can

formula

then

holds

uX e

L p/(n-l),

the

right

distributions have

if u G

L np/(n-p) loc

as

a diver-

and

side

the

of

in Ball

[5]

Since

]acobian

(5.&)

and

HI'P(n-I_< p< n)

order

n-i

defined

i.e.,

Ball,

if u E

of

is well

_< i,

(u 2 ,... ,u n) ..,x _1,x ..,x ) ~' ~-W n ~

(uX~(x

Hl'n(~ ;JR n).

if (n-l)/p+(n-p)/np

proved,

as

122-123)

~(Ul , 9 .. , U n) n ~ (5.~) det Du= ~(xl ' ,x )= z (-i) mn m=l ~x This

Du

in

belongs

the

p >_ n 2/(n+l).

Currie

and

sense Thus

Olver

to of we

[6 ], the

following result :

([5],[6])

5.1

LEMMA

p>_n2/(n+l) while

if

(5./~)

then

p >n,

holds.

det

det

Let

-

Du Du

is

Moreover,

if

of Hll'oc P

for p >n'/(n+l)

the sense

of distributions.

n = N>_2and

U@~ol(Q;~n).l_n (5.&)

is defined

by

defined

a L l1o c

as

u h converges , then

in

sider a function g(x, n) defined for x E ~ C ~ n a n d

in x a n d

(5.5)

and weak to

formula topology

det Du

in

functions, we connER

m with val-

(see in next section the reason to assume

g continuous

independent

of s) satisfying:

set {(X,n)

The

the

converges

To get a semicontinuity result for polyconvex

ues in [0,+~]

a distribution;

-function

to u

det Du h

as

If

: g(x,n) < + ~ } is

open

{and

not

empty)

with

respect

m

in

(5.6)

REMARK a

~

~ ~{

g(x, n) to

n E ~{

5.2

- Of

nontrivial

and

is m

g

convex

course,

condition only

is c o n t i n u o u s

and

the

lower

on

semicontinuous

semicontinuity

at the b o u n d a r y

19

this set.

assumption

in

(5.6)

is

of the d o m a i n

where

g

MARCELLINI

is finite. consider the

We

assume

(see

functions

described

also

the

f(x,Du)

det Du > 0, a n d

5-3

by

assuming

Ball by

go to + ~

the next l e m m a ,

but w e

(5.5),

have

Du)

[3] ),

that

if det Du

are

Thepe

functions

gk

in m i n d

in nonlinear

where

are

finite

if

could

elas-

considered

and

only

if

sequence

of

§ 0.

with two a p p r o x i m a t i o n

-

(5.6) w e

[5 ], of interest Antman

g(x,det

(De Giorgi)

peal nonne~ative

On

paper

=

Let us begin

LEMMA

to simplify

also other eases.

situation

ticity

(5.5)

exists

lemmas.

a

nondecr'easing

that conver'ge, as

k §

~ , to g. For m

every

k , g k ( X , n)

early

with

is uniformly continuous in ~

respect

to

n, it is convex

•

~

with respect

,it @ r O w S

lin-

q and

it is

to

equal to zero if dist ( x , a s

Proof. Let

equal

be g

to

is lower

one

of affine

Giorgt

[10]

a

ai_l,

9 : dist

and

argument of e a c h

gk

has

(x,~)

Oi ~ 0 . R

m

> 1/i},

and

every

x e~,

For

; thus

a.

point),

Let us

let

~i ~ C[(a)l

the

it is the s u p r e m u m

( a . ( x ) , n) + b . ( x ) . Like ] ] of [ 1 0 ] c a n be a p p l i e d ,

supporting in

gk(X, n) : m a x

sequence

on

functions

to be c o n t i n u o u s

(5.7)

The

on

(the

neighbour

b.(x) ]

{xE

semicontinuous

quence

in

ai =

we c a n

define

a0,

of a se-

in pag. since

function

31 of De

g is f i n i t e

choose

a.(x) ]

and

b0 -- O, a n d

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

all the required

properties.

r

LEMMA

5.4 - There

isfyin~

all

exists a sequence

the pvopepties

hk(X,n)

of C -functions sat-

stated in the previous

lemma

(except

the

fact that h k >- -i).

co

Proof. Let

be a mollifier,

i.e.

20

~ E C o ( ~ { n • ~9 m), I c~ d x d y

: 1,

MARCELLINI

c~ >0._ Let us define h k r : hk.~e,

where

If

s

is

is

zero

is

r +0

sufficiently if

dist

hkr

(x, ~a)_< 1/k+l.

h ke converges

r = e(k)

small,

such

a

c~e (x,n):e nonnegative

By t h e

to gk uniformly

-(n+m) a(x/e,n/e).

C -function,

uniform continuity m

in

a • ~

. Thus

of

we can

and

gk'

as

choose

that

1

(5.8)

For

lhk,e(k)

k >2

sequence verify

let

us

hk

that

define

satisfies h k is

hk

< h

the

= hk_l,e(k_l)

stated

increasing

with

1 +-'2-~

1 k-i

+--~

hk+l

prove

a

1 +-'i-k -r

+

1

1 k-I

~

semicontinuity

nalogous

result of Reshetnyak

THEOREM

5.5 -

example,

let

us

an

a-

to k :

-

1 _. k-1

1 k-I

hk+l

.

result,

that generalizes

[0,+,] of

v be functions

be a function

L 11oc(a;~m),

v h converges

to v in the sense of distributions,

(5.10)

f a (v h' ~)dx

lim

( k - l ) -1 . T h e

[28 ].

Let g: a • ~ { n +

(5.6) . Let v h a n d


_g(x,v

Since D g(x,v

We

= v

result

we

the

can

fact

for

use

that

> 7 g(x,v ) d x + / - ~ ~ c -~ O.

Fatou's D g is

In

(D g(x,v ),v-v )dx n ~ e

fact,

lemma, bounded

in

while in

a

the in

first term the

second

of the term

we

x R m independently

of

5.5 we obtain two semicontinuity

re-

n

By

lemma

5.1

sults for integrals

(5.14)

Here isfies

F(u)

:

and

theorem

of the type:

I n f(x,Du)dx

f(x, ~) is a polyconvex (5.5),

COROLLARY 5 . 6

Jacobians

(5.&)

Du h

function

det 2 Du .... )dx

like in (5.1), a n d

u he functions of

- Let u h a n d

and

Du

ape

holds

for

uh

topology of H~ocF(~;~N),1_

COROLLARY

I g ( x , D u , d e t Du,

g(x,n)

sat-

(5.6).

> rain {n2/(n+l);Nn/(n+l)}.

mula

=

Assume defined and

then

5.7 - Let u h a n d

that as

the L ~

loc

H1~7 ( R ; q R l r ~ N),

for

subdeterminants -functions

u, If u h converges

and

of

P

>

the

that for-

to u in the w e a k

lira inf F(u h) >F(u). h u be function

22

of Hl'r(loc~; ~ N)

. for r _>

MARCELLINI

>

min

If

uh

converges

{ n2/(n+l);

Nn/(n+l)

{n,N}.

for p >min

to u in

the w e a k

then

}

topology

lira i n f h

o f H)l o1c ( a. ; ~o N

F(u h) >F(u).

6. Some examples and remarks

Here

we

continuity let

us

able of

discuss

theorems

show

that

the

1.1 the

Let

us

result

does

not

hold

if g(x,

if g = g(x,s,n)

(with

to x ,

Of

course,

to

cases:

in

or

exhibit

fact,

if

theorem

5.5

in

topology

of L p .

weak

EXAMPLE

6.1

nonnegative, in

theorem

there

Let

-

n

:

bounded [19]

that

in

H1'2

a

exists

with

counterexamples,

reduces

usual

m = 1 and

let

g(x,

measurable for

every

sequence

in pe

uh

usual

must

= a(x)n

[1,+ -]

It and

2,

has

to

and

measur-

the

with

been

u

non se-

theorem

for every

converges

5.5

consider

semicontinuity

n)

semi-

meaning

some p > 1 ,

(0,1).

that

theorem

the

Inl p for

the

of t h e

n) i s o n l y

we

to

and

5 of

begin

g(x,n)>cost

micontinuity the

assumptions

5.5.

respect

coercive

of some

and

with

s).

necessity

in

a(x)

proved ueH 1'2 Lp

and

satisfies

1

(6.1)

1

lira I 0 a(x) h

(Uh)2dx

= 70 b(x)

(u')2dx

,

where

(6.2)

b(x)

=

lim

If w e

consider

mable

a.e.

and

(6.3)

v = u'.

lira h

in

2e[f

x+r

(y)dy]-1

X--E E+

a

0 +

function

(0,i),

Then,

Io~ v h

then

a(x) b(x)

for e v e r y

,dx

{

0

is

23

that

zero

a.e.

we

have

~>~ C7,

: - lim I~ h

a.e.

uh

r

Let

is us

not

locally

define

= -Io ~ u ~ ' d x

vh

sum= uh

: Io~ v ~ d x .

MARCELLINI

Moreover v h and if v is

not

(6.&)

our

6.2

In

Lipschitz

such the

that

fact,

continuous

the

function

wh = u~,

v~(x)

We c a n

-

product u(x)

L 1,

dx

w = u ~ = 1.

of

the

showed

is

product

Like

in

of

In

necessary.

We

(U~h,U~)

define

v h = (u~)',

[12]

a

in

u h converges

v h converges

to

sequence

defined

us

(0,1),

in

L'

to

v = (u~)'=

1,

to v i n

the

sense

L~ , b u t

cannot

I g(w(x),v(x))dx, with the

respect

L~

= i

extend

where to

norm

n,

topology

theorem

g(s,

and

n)

the

for

theorem

1.1

have

counterexampte

a

does

not converge

-

If

n

r < p

in

but

EXAMPLE 6 . 4

of

Eisen

exists

and

we

in

4.1

there

=-

, I o (wv) 2 d x

by

terexample

of

a.e.,

in

- costlsl np/(n-p),

sumption

since,

w,

is

5.5

inte-

continuous

topology and

to

the

in

considered topology

of

for v.

6,3

HI ' p ,

v~(x)dx

that

uh

(6.3),

v are

general

convex

distributions

is

(5.11),

= 0

Let

= 0

in

type

and

EXAMPLE

a(x)

counterexample

functions

v h and

that

( s , n), the

satisfy

1

means

grals

do n o t

fo

Eisen

u~(u~)'

I~ ( W h V h ) 2 d x

This

: O
_ 2 for the

integral

(6.7)

We

7ag(x,u(x))ldet

distinguish

fine

I nl ~

two

= + ~

tion a n d

Du(x)l a

cases:

if

in

the

~ _> 1 or

the integral

weak

topology

theorem

I.i in the general

with

a

nondecreasing

them

growing

pendent ous

function,

case

sequence

linearly

of s a n d

at

-),

continuous.

and

(~ < 0; in the s e c o n d c a s e

n < O. If g is a nonnegative

c~ >_ I, then

continuous

dx

either

(if a

~ > 1 or

n).

This

> 1 we can

of convex

If g

Carath6odory

func-

(6.7) is sequentially lower semi-

of H l , n ( ~ ; ~

and

we de-

from

functions

corollary

= g(x) ~ n2/(n+1):

(6.8) If

and

uh

weakly lira h

u are smooth

converges

inf

f~ g(x)l

{say

Uh'UE

Hl,n(~;~9 n))

and u h

to u in H l ' P ( a ; ~ n) , then det D U h la dx _> Jag(x)Idet DuJ a dx.

Let us mention explicitely that we

25

of

if g

then

5.7,

Inl ~

have

not proved that

(6.8)

MARCELLINI

is true

if l