Lower Semicontinuity in Spaces of Weakly Differentiable Functions

MaxPlanckInstitut

fur Mathematik

in den Naturwissenschaften

Leipzig

Lower semicontinuity in spaces of

weakly dierentiable functions

Jan Kristensen

PreprintNr

Lower semicontinuity in spaces of weakly

dierentiable functions

Jan Kristensen

Intro duction and the main results

In this pap er we adopt the approach initiated by Balder in for the study of multiple

integrals on Leb esgue spaces and use Young measures to obtain new lower semicontinuity

results for multiple integrals on BV R the space of functions of b ounded variation

The multiple integrals covered by the results are of the form

I u F x u ru dx

where R is an op en and b ounded set F x v X is a normal integrand and ru is

the density of the absolutely continuous part with resp ect to Leb esgue measure of the

distributional gradientof u

It is well known that the natural convexity assumption in the multidimensional calculus

of variations is quasiconvexityasintro duced by Morrey in Notation and denitions

are given in Section The classical lower semicontinuity results for multiple integrals

p n

dened on a Sob olev space W R cf and state that under suit

able growth conditions related to p quasiconvexity is a necessary and sucient condition

for sequential weak lower semicontinuity Without the growth conditions the lower semi

continuity results fail cf The growth conditions can be relaxed if one adopts the

approach prop osed in It amounts to redening I u for nonsmo oth u by a relaxa

tional pro cedure and is known as the Leb esgueSerrin extension of I u In the context

of quasiconvex integrands this programme was b egun in and there is bynownumer

ous pap ers on the sub ject Related to the study undertaken here are in particular

and where results on lower semicontinuity and relaxation were obtained for the

Leb esgueSerrin extension of multiple integrals dened on BV R Without certain

growth conditions the relaxational pro cedure dening the Leb esgueSerrin extension can

fail to provide an extension cf We refer to and for a systematic exp osition

and further references on quasiconvexityandlower semicontinuity

Unless otherwise sp ecied we assume throughout the pap er that m n and that

is an op en and b ounded prop er subset of R We dene I u for all relevant functions

Supp orted by the Danish Natural Science Research Council through grant no

Mathematical Institute University of Oxford England

J Kristensen

by the formula The integral in is understo o d as a Leb esgue integral or if

necessary as an upp er Leb esgue integral

The main results are contained in Theorems However we b elievethatsomeofthe

auxiliary results may b e of indep endentinterest in particular Lemma which contains

a result on truncation of sequences of gradients and Prop osition on approximation

of quasiconvex functions

Theorem p ertains to the case where u is of b ounded variation and has a distributional

gradient Du which is absolutely continuous with resp ect to a xed nonnegative and nite

Radon measure Following we dene for each p the space

dD u

p p n n

L W R u BV R Du

where dD ud denotes the RadonNikodym derivative of Du with resp ect to The

Sob olev spaces with resp ect to a measure enjoy the same compactness prop erties as the

usual Sob olev spaces cf In particular we note that if is a

nondecreasing function which satises the condition

as t

and if fu gW R satises

Z Z

dD u

sup ju j dx d

then for some subsequence of fu g for convenience not relab elled and some u W R

dD u dD u

weakly in L u u strongly in L and

loc

d d

n nm

Denition Let p An integrand F F x v X R R R b elongs

to the class I if it satises the following three conditions

H F F x v X is a normal integrand ie Borel measurable and lower semicontinu

ous in v X

H F x v X is quasiconvex in X for almost all x and all v

H For almost all x and all v

F x v X

if p lim sup

jX j

and no condition is required if p

It is not dicult to show see Lemma that under the hyp othesis H the condition

H is equivalent to the condition lim sup jF x v X jjX j p

Dening F inf fF gwehave the following result

Lower semicontinuity

Theorem Let be a nonnegative and nite Radon measure on p and

F I If fu gW R satises

p j

fF x u ru g is uniformly summable on

j j

and

sup kru k

j p

then

Z Z

F x u ru dx lim inf F x u ru dx

j j

Remark It follows from the pro of that if we replace the assumption with

dD u

sup

then wehave instead of the conclusion

Z Z

adx lim inf F x u adx F x u

a a

where a ddL

The lower semicontinuity prop erties of multiple integrals with quasiconvex integrands

have previously been studied in this setting by Ambrosio Buttazzo and Fonseca in

m s m s

Write a L and Du ru L D u for the Leb esgueRadonNiko dym

decomp ositions with resp ect to Leb esgue measure and dene the functional

Z Z

ru dD u

E u adx d F x u Gx

a d

Then the principal result of guarantees lower semicontinuityof E u on sequences fu g

satisfying and when F is a Caratheo dory integrand satisfying H and H

G is a rank convex normal integrand p and a L It is p ossible to relax

the conditions on G see and we remark that the singular part in E u also is lower

semicontinuous under the conditions of Theorem Observe that these results easily

give existence results for minimisation problems where the discontinuity set is imp osed a

priori

The pro of in is achieved by considering the absolutely continuous part and the singular

part of E u separately The singular part of E u is then treated by use of a result from

In dealing with the absolutely continuous part the authors use a result from on

Lusintyp e approximation of functions of b ounded variation byLipschitz functions Such

approximation results were rst established for Sob olev functions in and and used

in to obtain lower semicontinuity results for multiple integrals on Sob olev spaces

The main novelty of our result is that we allow p and a dD udL L

The extension from a L to a L app ears to b e essential for the pro of of our second

lower semicontinuity result stated in Theorem Apparently this extension also requires

J Kristensen

a dierent strategy for the pro of as the metho d based on approximation with Lipschitz

functions seems to break down if a is not essentially b ounded We also note that the reason

that we can allow the integrand F to b e merely normal and not necessarily Caratheo dory

is due to our approachviaYoung measures

Before stating Theorem we recall that if u is of b ounded variation then it is p ossible

to dene a measure theoretic normal N to the jump set S for u and to dene onesided

u u

m s

traces u u of u on S see Section Let Du ru L D u be the Leb esgue

RadonNiko dym decomp osition of Du with resp ect to Leb esgue measure The space

n n s s

SBV R fu BV R D u D ubS g

of sp ecial functions of b ounded variation and its generalisation GS B V R see Sec

tions and were intro duced byAmbrosio and De Giorgi in as a natural setting for

weak formulations of free discontinuity problems

Theorem concerns the case where u is a sp ecial function of b ounded variation or

more generally lies in GS B V R and is motivated by a compactness result due to

Ambrosio cf Theorem The compactness result can be stated in the following

manner If is as in if is concave nondecreasing and satises

as t

and if fu gSBV R satises

Z Z

sup ju j jru j dx ju u j dH

j j

then for some subsequence of fu g for convenience not relab elled and some u GS B V R

u u in measure ru ru weakly in L

j j

j dH u ju and sup

j j

For convenience we state a precise version of the compactness theorem for GSBV in

Section

Theorem Let satisfy p and F I If fu gSBV R satises

p j

fF x u ru g is uniformly summable on

j j

and

sup kru k

j p

then

Z Z

F x u ru dx lim inf F x u ru dx

j j

Lower semicontinuity

Remark It is p ossible to give a more general version of Theorem in terms of GSBV

functions we indicate how to do this in Section

Lower semicontinuityofmultiple integrals with quasiconvex integrands has b een studied

in this setting byAmbrosio in under the assumption t The main novelty

of our result is the extension to the case when lim t eg t t where

Wecover exactly the cases describ ed in the compactness theorem for GSBV

Consider the functional

Z Z

E u F x u ru dx Gx u u N dH

If for some constants c c c andwehave

F x v X c jv j c jX j and Gx u v N minfc ju v j g

then by the compactness theorem the functional E is co ercive in the space GS B V R

Under the conditions of Theorem the bulk energy term of E u is lower semicontinu

ous Corresp onding lower semicontinuity results for the surface energy in E uhave b een

obtained by Ambrosio in for the cases However under the condition that G is

regularly biconvex the metho ds of also yield lower semicontinuity of the surface energy

in the case see also the remark following Theorem We also notice that E u

can be lower semicontinuous even though the bulk energy and surface energy are not so

separately see and the references therein

As already mentioned we pro ceed as suggested by Balder in and establish the lower

semicontinuity results of Theorems and by use of Young measures By virtue of

the hyp otheses of either Theorem or Theorem wecanby extracting a subsequence

if necessary assume that fru g generates a Young measure dx see Section

j x x

for notation According to a general result from see also Theorem

Z Z

lim inf I u F x uxX d X dx

j x

Since ru ru weakly in L it follows that almost all haveacentre of mass denoted

j x

and that ruxalmosteverywhere If therefore wehave by

x x

ae F x uxX d X F x ux

x x

then the lower semicontinuity results of Theorems and follow We express

by saying that Jensens inequality holds for F x ux and for almost every x The

key feature of this approach is that it allows us to ignore the x v dep endence in F We

establish and hence Theorems and in the following manner Let Q denote

the class of quasiconvex functions f R R satisfying the growth condition

f X

lim sup

jX j

if p no condition is required if p Note that by denition of the class I of

admissible integrands the functions f X F x uxX b elong to Q for almost all x

x p

J Kristensen

The main part of the pro of aims at establishing that for almost all x

f Q fd f

x p x

A result due to Kinderlehrer and Pedregal states that for a xed x is

equivalent to the requirementthatL b b e a gradient pYoung measure Theorems

and establish exactly this Before pro ceeding further a few remarks are in order

The semicontinuity result was obtained indep endently by Pedregal for the

case of Caratheo dory integrands that are bounded from below Pedregal also emphas

ised the imp ortance of Jensens inequality in problems of lower semicontinuityand

highlighted it in his denition of closed W quasiconvexity This approachwas also used

in and in to obtain lower semicontinuity results for multiple integrals on W

The go o d lo calisation prop erties of Young measures were also used in to study the

principle of convergence of energies

The key results of this pap er are the following two theorems

Theorem Let be a nonnegative nite Radon measure on and let fu g be a

sequenceinBV R satisfying and If fru g generates the Young measure

m m

dx then for L almost al l x L b is a gradient pYoung measure

x x x

Theorem Let fu g be a sequence in SBV R satisfying and If

m m

fru g generates the Young measure dx then for L almost al l x L b

j x x

is a gradient pYoung measure

Remark We present a more general statement of Theorem in terms of GSBV functions

in Section

In some sense Theorems and are surprising Quasiconvexity is dened with

sp ecic reference to gradients but a result of Alb erti states that the approximate

gradient ru of an SBV function u can be any summable function V R

Of course it is the conditions we imp ose on the measures Du that force the sequence

fru g to generate a Young measure with the ab ove prop erty One might think that it

should be p ossible to decomp ose ru into a gradient and another term that converges

stronglysothat the Young measure is essentially generated by the gradients Of course

if p ossible this would prove the theorems However by Example this approach

cannot be successful Example displays a sequence fu g satisfying simultaneously

all the conditions in Theorems but where the sequence fru g of approximate

gradients and any subsequence thereof do es not admit a decomp osition of the form

ru E rv where fE g converges strongly in L R and fv g weakly in

j j j j j

W R

Theorems and are very close to b eing optimal In Section we present examples

showing that if one of the conditions or is slightly weakened the

corresp onding conclusion is false

The pro of of Theorem is presented in Section By use of a truncation argument

and the VitaliHahnSaks Theorem we deduce in Section Theorem from Theorem

Lower semicontinuity

As noticed ab ove Theorems and are then easy consequences via and

Toprove Theorem we exploit the characterisation of gradientYoung measures due to

Kinderlehrer and Pedregal see also or contains a generalisation

We only need the following sp ecial case of their result

Lemma D Kinderlehrer and P Pedregal special case of Suppose is a

probability measure on R satisfying

jX j d X

The homogeneous Young measure L b is a gradient Young measure if and only

if for al l quasiconvex functions f for which f X jX j as jX j the fol lowing

inequality holds

f X d X f

In the App endix we present an almost selfcontained pro of of the full characterisation

of gradient pYoung measures covering all cases p We obtain at the same time

a slight renement of the results in in the sense that we are able to show that it

is only necessary to test in with quasiconvex functions that equal jX j outside large

balls

As a rst step towards proving Theorem we employ Lemma to show that for

almost all x the measure L b is a gradientYoung measure We establish

by use of wellknown results on dierentiation of measures along with the following result

Lemma Let f R R be a nonnegative quasiconvex function satisfying

f X jX j as X and let u R be of local ly bounded variation For

n nm

x r dist x a R and X R the fol lowing inequality

holds

Z Z

f ru dy jD ujB juy a X y j dy

B B nB

xr xr

x r

L B f X

x r

This lemma is reminiscent of Lemma in Wederive it as a corollary of a slightly

more general inequality in Section the pro of of which is elementary A similar result

app ears to b e false for quasiconvex functions with sup erlinear growth at innity

To conclude the pro of of Theorem weinvoke the following technical result on trunca

tion of sequences of gradients More precisely the conclusion is established using Corollary

stating in particular that a gradient Young measure is a gradient pYoung measure

order moment if and only if it has a nite p

J Kristensen

Lemma Assume that is a bounded Lipschitz domain Let fu g satisfy u u

j j

p m

weakly in W Assume that for some p thereisasequence fV gL R

with the properties

if p fjV j g is uniformly summable on

if p fV g is bounded in L R

and

ru V in measure on

j j

Then there is fv gC such that

v weakly weakly if p in W

kru ru rv k

j j

and if p

fjru rv j g is uniformly summable on

The case p was treated by Zhang in An elementary pro of for the cases p

relying on the Ho dge decomp osition is given in Section We apply the result to vector

valued functions by applying it to each co ordinate function The fact that the result

is scalar ie it is p ossible to prove it for realvalued functions and then transfer it to

vectorvalued functions by applying it to each co ordinate function paves the way for many

dierent extensions and pro ofs see

Using this result we easily derive two useful corollaries The rst concludes the pro of

of Theorem and concerns the p ossibilityof nding generating sequences for gradient

Young measures with go o d integrability prop erties

Corollary Let dx bea gradient Young measure and let u W R

x x

be an underlying deformation Let p and assume that has a nite p order mo

ment ie if p

Z Z

jX j d X dx

and if p there is a compact set C R such that for almost al l x the measure

is carried by C

R such that v weakly weakly if p in Then there exists fv g C

j j

n p

W R fru rv g generates and if p the sequence fjru rv j g is

j j

uniformly summable on

Corollary is a slight renement of a similar result in where the pro of was based

on a stability result from Other results in the same vein have also b een obtained in

using arguments based on the Lusintyp e approximation of general Sob olev functions

with Lipschitz functions as in The rst result of this kind seems to come from

and was obtained in an indirect way Recently similar results have been obtained in

within the more general setting of comp ensated compactness

The second corollary to Lemma is the following

Lower semicontinuity

Corollary J Bal l and K Zhang Suppose f R R is a quasiconvex

function satisfying

f X

if p lim sup

jX j

and no growth condition if p Let dx be a gradient pYoung measure

x x

Then

fd f

x x

holds for almost al l x

The pro of in is based on the lower semicontinuity result in and Chacons Biting

Lemma

By virtue of it is clear that Theorems and follow from Theorems

and Corollary

It is p ossible to prove Theorems and in the case p without relying on

the characterisation of gradientYoung measures The pro of then relies on Lemma and

the following approximation result which again is proved using Lemma Corollaries

and

Prop osition Let f R R be a quasiconvex function such that for some c

c and p

p p

c jX j c f X c jX j

holds for al l X Then there exist f R R that are quasiconvex and satisfy

a f X f X

j j

b f X f X as j

c there exist a r b R such that

j j j

f X f X a jX j b if jX jr

j j j j

where f denotes the convex envelope of f

The approximation result in implies the existence of quasiconvex functions g satis

fying a b and

c there exist r suchthat g X c jX j c if jX jr

j j j

The pro of in is based on a result on higher integrability of certain minimising se

quences For our purp oses it is imp ortantthatwehave c and not c

We note that assumption cannot be avoided In Example we show by use

of results from and that the p olyconvex function f X jX j jdet X j where

p dened on R cannot b e approximated from b elowby subquadratic rank

convex functions

J Kristensen

The pap er is organised as follows In Section we briey recall the main denitions

and state some preliminary results Section is devoted to proving the Decomp osition

Lemma Lemma and its corollaries In Section we give the pro of of Prop osition

The main results Theorems are proved in Sections and In Section we

have gathered some examples that illustrate the sharpness of our hyp otheses Section

is an app endix and contains an essentially selfcontained pro of of the characterisation of

gradientYoung measures

The present pap er is a revised and extended version of an earlier manuscript where the

most imp ortantchanges are that Prop osition and the full pro of of the characterisation

of gradientYoung measures have b een included The result stated in Theorems and

together with a pro of based on was announced at the workshop Calculus of Variations

and Nonlinear Elasticity in Cortona Italy June

Notation and preliminary results

In this section we gather some denitions and elementary results that are used in the

sequel

Basic notation

Our main references for measure theory are and Except for the Hausdor

measure H all measures o ccurring in this pap er are Radon measures If is a measure

and A is a set then the measure bA is dened as bAB A B

Let O be either an op en or a compact subset of R and let BO denote the eld

of Borel subsets For a b ounded R valued Radon measure on O the total variation on

A BO is

jjAsup jA j

where the supremum is taken over all partitions of A into countably manyBorel subsets

A and jA j denotes the usual Euclidean norm of A The function jj is called the

i i i

total variation measure for and is a nonnegative nite Radon measure on O

As the concepts of uniform absolute continuity and uniform summabilityarecentral to

the presentwork we display a formal denition

Denition Let beanonnegative nite Radon measureonO Afamily of R valued

bounded Radon measures on O is said to be uniformly absolutely continuous with respect

to briey uniformly AC if for any there exists a such that for B BO

B jB j for al l

A family F of summable functions V O R is uniformly summable if the family

fV V F g is uniformly AC

Lower semicontinuity

Remarks By the RadonNiko dym Theorem is uniformly AC if and only if each

measure in is absolutely continuous with resp ect to and the family F fdd

g of RadonNiko dym derivatives is uniformly summable

If has no atoms and f g is uniformly AC then sup j jO

j j

f g is uniformly AC ifandonlyifthesequence fj jg of total variation measures is

j j

uniformly AC

k d d k

Denote by C O R the space of R valued C functions on O The subspace of func

k d k

tions with a compact supp ort is denoted by C O R If d we simply write C O

instead similarly for all other function spaces

d d

Denote by C O R the space of R valued continuous functions with the prop erty

for every there is a compact set K O such that jxj if x O n K Of

d d

course if O is compact C O R C O R Endowed with the supremum norm

denoted by kk C O R is a separable Banach space By the Riesz Theorem the

dual space C O R can by the duality pairing

x djjx h i x

djj

be identied with the space of b ounded R valued Radon measures on O The corres

p onding norm of is kk jjO

Our reference for approximate limits and derivatives is Let O be an op en subset

D d d

of R and u O R a Borel function We take S R fg to be the one point

compactication of R and consider u as a function with values in S Let d b e a compatible

metric on S Take F BO and x O with the prop erty L F B for all

x r

r Here and throughout the pap er B denotes the op en ball centred at x with

x r

radius r

Approximate limit v S is said to b e an approximate limit in x for u in the domain F

written

ux v ap lim

x x

x F

lim duxv dx

B F

x r

The approximate limit is unique if it exists

Jump set The jump set S of u is dened as the set of p oints where u has no approximate

limit ie

S fx O ap lim uy do es not existg

y x

y O

The set S is Borel L S and

u u

ux ap lim uy

y x

y O

for almost all x O n S In case holds at x wesay that u is approximately continuous

at x

J Kristensen

Approximate gradients The function u O R is approximately dierentiable at x

if it is approximately continuous at x and if there exists a matrix X R suchthat

jux ux X x x j

ap lim

x x

jx x j

x O

In case u is approximately dierentiable at x the matrix X in is uniquely determined

and is called the approximate gradient of u at x It is denoted by rux The set

of points r where u is approximately dierentiable is a Borel set and the function

ru r R is a Borel function

Our references for function spaces are and for BV and SBV We use the notations

kuk kuk kuk kuk

p d p d

pO pO

L O R W O R

and if O R weomitO from the notation The distributional gradient of a distribution

u is denoted by Du In particular if u O R is of b ounded variation then Du

fu x g is a R valued b ounded Radon measure on O Furthermore in this case u

r s

is approximately dierentiable almost everywhere and

juy ux ruxy xj dy as r

L almost everywhere The Leb esgueRadonNiko dym decomp osition of Du takes the

form

D s

Du ru L D u

where D u is a singular measure The jump set S of u is countably D rectiable

S K N

u i

where H N and K are compact sets each contained in an emb edded C hyp er

surface There exists a Borel function N S R such that jN xj for all x

i u u u

and suchthat N x is normal H almost everywhere in K to the surface We refer

u i i

to N as a unit normal to S it is clearly not unique

u u

If corresp onding to a unit vector N and a p oint x we dene the halfspaces

x N fy R y x N g

then the approximate limits

u x N x ap lim uy u x N x ap lim uy

u u

y x y x

y x N x y x N x

u u

exist for H almost all x S We suppress the dep endence on N and write simply

u u

u xandu u x Notice that the matrix u u N is uniquely determined u u

Lower semicontinuity

D s

H almost everywhere on S The Leb esgue decomp osition of D u with resp ect to

D ubS is

s D

D u Cu u u N H bS

u u

where Cu is a singular measure suchthat jCujA when A BO andH A

A function u O R is a sp ecial function of b ounded variation briey u

d s s

SBV O R if u is of b ounded variation and Cu or equivalently D u D ubS

Integrands and Young measures

Denition LC Young A Young measure on R is a nonnegative Radon

d d m

measure on R with the property A R L A for al l Borel subsets A of

Remark The denition of Young measure used here follows that of Berlio cchi and Lasry

It can b e shown to b e equivalent to the original denition due to Young and the ones

used in eg and

d m

Notice that a pro duct measure on R of the form L b is a Young measure

d d

on R exactly if is a probability measure on R SuchYoung measures are called

homogeneous Often it is clear from the context that all Young measures considered are

Young measures on some sp ecic set and in such cases we simply sp eak of Young measures

Denition An elementary Young measure is a Young measure for which there exists

m d

a L measurable mapping V R such that

Z Z

fd f x V x dx

for al l f C R

Remark If is an elementary Young measure as ab ove then we write

V x

where is the Dirac measure on concentrated at x and is the Dirac measure on

V x

R concentrated at V x

Prop osition Proposition pp Let bea Young measureon R

Then there exists a mapping x from into C R f g the set of

nonnegative nite Radon measures on R with the fol lowing properties

i For any Borel function f R the function x f x X d X is

L measurable and

Z Z Z

fd f x X d X dx

d d

R R

d m

ii R for L almost al l x

Furthermore if x is another such mapping then for L almost al l x

x x

J Kristensen

Remarks Our Prop osition is a sp ecial case of Prop osition in

We summarise the contentofiby writing dx

x x

Let fV g b e a sequence of measurable mappings of into R Since the corresp onding

sequence f g of elementary Young measures is always b ounded in C R it follows

from Alaouglus Compactness Theorem that there is a subsequence fV g and a measure

C R such that

weakly in C R

The following lemma characterises the case where is a Young measure

Lemma N Hungerbuhler Kristensen Under the above assumptions the

measure is a Young measure if and only if

sup L fx jV xjtg as t

The condition is equivalent to the fol lowing condition there exists a Borel function

h R such that hX as X and

sup hV dx

Remark In case and hold wesay that the sequence fV g generates the Young

measure

The next lemma is wellknown and is easily proved using Lemma

Lemma Let fV g fW g be two sequences of measurable mappings of into R If

j j

fV g generates the Young measure and if V W in measure then also fW g

j j j j

generates the Young measure

Denition An extendedrealvalued function F R R fg is cal led a normal

integrand if F x v everywhere if F is Borel measurable and if for every xed

x the partial function F x R R fg is lower semicontinuous

Denition Arealvalued function F R R is cal led a Caratheodory integrand if

both F and F are normal integrands

In the statement of the next theorem we use the notation F inf fF g

Theorem Let fV g be a sequence of measurable mappings of into R and assume

that it generates the Young measure If F R R fg is a normal integrand

and if fF V g is uniformly summable then

Z Z

lim inf F x V x dx Fd

Lower semicontinuity

If additional ly F is a Caratheodory integrand then fF V g is uniformly summable on

if and only if

Z Z

lim F x V x dx Fd

Proof The pro of can b e obtained by mimicking the pro of of Theorem in The rst

part of the theorem and the if part of the second assertion are proved in and in

The only if part of the second assertion of the theorem is proved in

Denition D Kinderlehrer and PPedregal Let p AYoung measure

nm p n

on R is a gradient pYoung measureifthere exists a sequence fu g in W R

such that

p n

i fu g is weakly weakly if p convergent in W R

ii weakly in C R

Remark We call the limit u of fu g an underlying deformation for It follows from

Theorem that if dx then for almost all x the probability measure

x x x

and has a centre of mass

rux

We end this subsection with an elementary but very useful observation Supp ose that

fV g generates the Young measure dx and that V R is a measurable

j x x

mapping Then the sequence fV V g generates a Young measure

x x

V x

where denotes the convolution of the two measures and dened as

x x

V x V x

h f i f v V xd v f C R

x x

V x

Observethatconvolution with simply corresp onds to a translation with V x

V x

Quasiconvexity

Denition CB Morrey A function f R R fg is quasiconvex at

nm m m

X R if for every open and bounded set R with L one has

Z Z

f X rux dx f X dx L f X

for al l u W R for which the integral on the left hand side exists The function

f is quasiconvex if it is quasiconvex at every X R

Remarks It is enough to know that holds for one nonempty op en and b ounded

set Indeed if holds for one op en and b ounded set not necessarily with

J Kristensen

m m

L then it holds for all op en and b ounded sets with L If f

then the condition that L can b e omitted see eg

It can be shown see eg that a realvalued quasiconvex function is rank

convex ie f is convex on rank lines in R

A function f R R fg is separately convex if it is convex on lines parallel

to the co ordinate axes It can be shown that a realvalued separately convex function

is lo cally Lipschitz continuous see Furthermore we have the following elementary

lemma

Lemma Suppose that f R R is separately convex and that for some p

f X

lim sup

jX j

Then also

jf X j

lim sup

jX j

Proof For R the following inequality holds

mn mn

f inf f X sup f X

jX jR

The lemma follows from this To derive we take X such that jX j R and

f X inf f X Let X X X denote the orbit of X under reections

jX jR

in the co ordinate hyp erplanes it is not assumed that the X s are distinct By separate

convexity

mn i

f f X

and since jX jR inequality follows

Denition Let f R R fg be an extended realvalued function The

quasiconvex envelope f of f is dened as

f X supfg X g quasiconvex and g f g

Remark It is not excluded that f

Lemma D Kinderlehrer and P Pedregal the appendix Let f R R

m m

beacontinuous function Let an open and boundedset R with L begiven

and dene the extended realvalued function Q f R R fg as

Q f X inf f X rux dx u W R

Then Q f f

Lower semicontinuity

Lemma B Yan Lemma Let f R R be a continuous function

n nm

and let B fx R jxj g For any X R there is a sequence fu g in

W B R such that

f X ru dx f X and ku k

j j B

For quasiconvex functions of linear growth at innitywe have the following elementary

lemma which seems to have been overlo oked in the literature

Lemma Let f R R be a quasiconvex function satisfying

jf X j

lim sup

jX j

and dene f X lim sup f tX t Let be a bounded Lipschitz domain in R

n n

and denote by N the outward unit normal on For u W R a R and

X R the inequality

Z Z

m m

f ru dx f a Xx ux N x dH x L f X

holds

Remarks The recession function f is a p ositively homogeneous Lipschitz continu

ous function

R R

By the Divergence Theorem rudx u N dH

Proof It suces to prove the assertion for a and X Since is a Lipschitz

m n

domain we can assume that u W R R Let f g be a standard C mollier

t t

and dene for the convolutions

and u u

Observe that outside fx distx g and consequently since f is

quasiconvex at

L f f ru u r dx

Wenow employ an auxiliary function as in For each x weintro duce the p ositively

homogeneous Lipschitz function

f xru xtX f xru x

g X sup

By rank convexityoff g X f X whenever rankX and therefore

f ru u r f ru f u r

J Kristensen

Consequently

Z Z

L f f ru dx f u r dx

and therefore letting we obtain by virtue of Reshetnyaks Continuity Theorem

see Theorem

Z Z

m m

L f f ru dx f u N dH

Finally we conclude by letting and applying eg Theorem of

Proof of Lemma Let f g be a standard C mollier and put u u Note

that if is suciently small then u is welldened and smo oth on B Because

the function t ju y a Xyj dH y is continuous on rr we can nd

R rr suchthat

Z Z

ju y a Xyj dy ju y a Xyj dH y

B nB B

x r

If we apply with B noticethatf X jX j and make appropriate use of

f we get with u in place of u We conclude the pro of by letting tend to

and using Reshetnyaks Continuity Theorem see Theorem

It is also p ossible to prove directly by means of an argument which is similar to

the pro of of Lemma in

Pro of of the Decomp osition Lemma

In this section weprove Lemma and its corollaries The pro of of Lemma in the

case p is obtained in three steps each stated as a lemma The main to ol for the pro of

is the Ho dge decomp osition which is used in the last step The case p is as noted in

the Intro duction a result due to Zhang

Lemma Suppose that is a bounded Lipschitz domain Let p let fu g be

p p

a sequence which converges weakly to in W and for which fjDu j g is uniformly

summable on Then there exists a sequence fv g in C such that

u v strongly in W

j j

Proof To simplify notation we assume that fx jxj g The pro of in the general

case is analogous By virtue of the RellichKondrachov Compactness Theorem

ku k

j p j

Lower semicontinuity

Take a cuto functions compactly supp orted in and with on B

j j

and Lip Dene w u Clearly w W w strongly in L

j j j j j j j

and

ku k krw ru k jru j dx

j p j j p j

It follows that u w strongly in W Because C is dense in W the

j j

assertion of the lemma follows

I am indebted to Stefan Muller for bringing the following result to my attention

p m

Lemma Let p and let fV g beasequenceinL R Then the fol lowing

three assertions are equivalent

a fjV j g is uniformly summable on

q m

b For al l qp and al l there exist W L R such that

sup kW k and kV W k

j q j j p

q m

c For some qp and al l there exist W L R such that

sup kW k and kV W k

j q j j p

Proof It follows easily by writing down the denitions

Lemma Suppose that is a bounded Lipschitz domain and that u weakly in

W Assume furthermore that for some p there is a sequence fV g of vector

p m p

elds in L with the properties V almost everywhere on R n fjV j g uniformly

j j

summable on and ru V in measureon Then there exists a sequence fv g in

j j j

C such that

kv u k and fjrv j g is uniformly summable on

j j j

Before weembark on the pro of we recall some facts on the Ho dge decomp osition

p m m

Let L denote the Leb esgue space of all psummable vector elds V R R and con

p p

sider the subspaces K and H consisting of resp ectively the curlfree and the divergence

free vector elds ie

p p p p

K fV L curlV g and H fV L divV g

p p p m p

It is readily seen that K and H are closed in L If u W R and ru L

loc

p p

then clearly ru K and it is not dicult to show that all vector elds in K can be

represented this way ie

p p

K fru u L and ru L g loc

J Kristensen

The orthogonal complementofK in L is H Therefore L K H by the Pro jection

Theorem Let K and H denote the corresp onding orthogonal pro jections We extend the

Hodge decomposition V KV HV to all V L byshowing that the op erators K

and H are of strong typ e p pforallp

The orthogonal pro jection of L onto K coincides with the metric pro jection onto K

Therefore KV ru where u L R is a minimiser of the functional

V V

jru V j dx u L R kru V k

By convexity it follows that u is the unique up to additive constants solution of the

EulerLagrange equation

divDu divV

Using the Fourier transformation denoted by Fwe derive the formula

KV F M FV

where M j j In view of the Mihlin Multiplier Theorem it follows that K

is of strong typ e p p for all p and of weak typ e

Proof of Lemma Only the case p requires a pro of By Lemma we can assume

p m

that u W and therefore we may extend each u to a function in W R by

j j

dening u outside

From the DunfordPettis Theorem we infer that fru g is uniformly summable on

Also fV g is uniformly summable on hence applying Vitalis Convergence Theorem to

the sequence fru V g on we deduce kru V k Since ru V outside

j j j j j j

kru V k

j j

By there are w L with rw L and w such that rw KV

j j j j j

loc

Because sup kV k it follows from that V weakly in L and since K is

j p j

of strong typ e p p rw weakly in L Therefore in view of Poincares inequality

w weakly in W Since ru rw ru V HV andHV HV ru

j j j j j j j j j

we infer from and the weak typ e of H that in particular

ru rw in measure

j j

This together with uniform summability on implies by Vitalis Convergence Theorem

that kru rw k By the RellichKondrachov Compactness Theorem we deduce

j j

that ku w k

j j

Next we show that fjrw j g is uniformly summable on Let q p be xed Take

Since fjV j g is uniformly summable on we can by Lemma nd W verifying

j j

kV W k and sup kW k

j j p j q

Recall that V outside and dene likewise W outside Then clearly kV

j j j

W k and consequently krw KW k c and sup kKW k c sup kW k

j p j j p p j q q j q

j j

By Lemma we infer that fjrw j g is uniformly summable on

The pro of is concluded by use of Lemma

Lower semicontinuity

Lemma K Zhang Lemma Suppose that is a bounded Lipschitz domain

and that u weakly in W Assume furthermore that there is a sequence fV g of

j j

vector elds in L with the properties V almost everywhereonR n sup kV k

j j

and Du V in measureon Then there exists a sequence fv g in C such

j j j

that

kv u k and sup kv k

j j j

Remark This statement is not identical to Lemma in but follows easily from its

pro of Some subtle generalisations have b een obtained recently in

Proof of Lemma The case p is covered by Lemma and the case p by

Lemma

With Lemma at our disp osal Corollaries and are easy to prove

p n

Proof of Corol lary It is clear that the underlying deformation u b elongs to W R

and b ecause can b e written as an increasing union of Lipschitz domains each compactly

contained in we can assume that is a Lipschitz domain

By assumption we may nd a sequence fu g such that u u weakly in W R

j j

and fru g generates Because fru rug generates the Young measure

j j x

dx we can without loss of generality assume that u

rux

Supp ose rst that p Dene for t the mapping T X minft jX jgXjX j and

notice that the sequence fjT ru j g is uniformly b ounded By Theorem it follows

t j

that

Z Z

p p

lim jT ru j dx jT X j d x X

t j t

and thus

Z Z

p p

jX j d x X lim lim jT ru j dx

t j

We can therefore nd t such that

Z Z

p p

lim jT ru j dx jX j d x X

t j

Let V T ru and notice that V ru in measure on By Lemma fV g

j t j j j j

generates the Young measure and therefore in view of and Theorem fjV j g

is uniformly summable on We conclude by applying Lemma to each rowinfru g

and by utilising Lemma once again

Assume next that p Then there exists a R such that for almost all x the

supp ort of is contained in the ball jX jR Thus if we take tR and let V T ru

x j t j

wehave

lim sup jru V j dx

j j

and therefore the claim follows if we apply Lemma to eachrow

Proof of Corol lary Assume that p The cases p f g are left to the

interested reader

J Kristensen

Let u denote an underlying deformation for Take a ball B and dene the

measure as

y y

ruy

Clearly is a gradient pYoung measure on B R which has v as an

underlying deformation Consequently there is a sequence fv g C B R as in

j xr

Corollary Due to the growth condition the sequence ff ruxrv g is

uniformly summable on B and therefore we obtain by Theorem and the denition

Z Z Z

lim f ruxrv dy f X rux ruy d X dy

j y

B B

xr xr

By quasiconvexitywehave for all j

f ruxrv dy L B f rux

j xr

and the conclusion nowfollows from Leb esgues Dierentiation Theorem

Approximation of quasiconvex functions

This section contains a pro of of Prop osition For the purp ose of proving the lower

semicontinuity results of Theorems and in the case p the imp ortantfact

is that the approximating functions can be taken to be of linear growth at innity The

renement that they can b e taken convex outside large balls is only used in the App endix

The pro of is divided into two steps each formulated as a lemma In the rst step it is

shown by use of Corollaries and that it is p ossible to approximate with sp ecial

quasiconvex functions of linear growth at innity The next step uses Lemma and

concerns approximation of the sp ecial quasiconvex functions encountered in the rst step

The desired approximation result follows from this

Lemma Let f R R bea quasiconvex function such that for some c c

and p

p p

c jX j c f X c jX j

holds for al l X Then there exist f R R that are quasiconvex and satisfy

a f X f X

j j

b f X f X

c lim f X jX j R for each j

X j

Lower semicontinuity

Proof We dene f as f F where F X minfj jX jfX g Notice that

j j j

hereby f are quasiconvex and satisfy conditions a and c It is also clear that f f

j j

By translation it suces to show that f f For that purp ose we apply Lemma

B R where B fx jxj g such that ku k and and nd u W

j B j

F ru x dx f

j j j

Let t R denote the righthand side of Extracting a subsequence if necessary we

can assume that t t R The lower bound in implies that fru g is b ounded

j j

in L and hence that fu g is b ounded in W B R By Theorem we can extract a

subsequence for convenience not relab elled such that fru g generates a Young measure

on B R

Put E fx B j jru xj fru xg and notice that

j j j

Z Z

t L B j jru j dx f ru dx

j j j

E B nE

j j

and therefore by and the lower b ound in

Z Z

jru j dx and sup jru j dx

j j

E B nE

j j

By de la ValleePoussins criterion fru g is uniformly summable on B and since

B nE

jru j strongly in L B itfollows that fru g is uniformly summable on B This

j E j

implies that u weakly in W B R and that is a gradient Young measure

Passing to the limit in yields by Theorem applied to the normal integrand

ru F t X t f X and the sequence

B nE

Z Z

f ru dx lim f L B lim inf f X d x X

j j

B R B nE

Referring to the lower b ound in it follows that has a nite p order moment and

therefore by Corollary that it is a gradient pYoung measure The pro of is concluded

using Corollary taking into account that an underlying deformation for is u

Lemma Let f R R be a quasiconvex function such that

f X

lim

jX j

There exist f R R that are quasiconvex and satisfy

a f X f X

j j

b f X f X

c f X f X a jX j b for large jX j where a b R

j j j j j j

J Kristensen

Proof Since f is b ounded from b elowwe can without loss of generality assume that f is

strictly p ositive Fix and take c c Rsuch that

f X jX j c

for all X For each p ositiveinteger k we dene the function g R R as g G

k k

where

f X if jX jk

G X

jX j c if jX j k

Observe that g is quasiconvex satises g X jX j c if jX j k and g X

k k k

g X f X for all X and all k

Fix X R We claim that

f X lim g X

Since g X inff G X ru dx u W B R g and G X jX j c for all

k k k

X and k there exists by Lemma a sequence fu gW B R satisfying

g X L B G X ru dx

k k k

u strongly in L B R and sup jru j dx

k k

If E fx B jX ru xj kg then

k k

Z Z

lim g X L B lim sup f X ru dx lim inf jX ru j dx

k k k

k k

B E

R R

Since lim inf jX ru j dx lim sup f X ru dx and since by Lemma

k k k

E B

it follows that

f X ru dx f X L B lim sup

the inequality follows

Next take for a p ositiveinteger j j and consider the corresp onding sequence

fg g as constructed ab ove For each j there exists by and Dinis Lemma an

integer k suchthat

f X if jX jj g X

If we dene f max fg i jg then f are quasiconvex and satisfy ac

j j

Indeed a and b are obvious from the construction and c follows b ecause

f X max f ijX j c g

j i

with equalityifjX jmax k

ij i

Proof of Proposition This is a straightforward consequence of Lemma and Lemma

Lower semicontinuity

Lower semicontinuity in Sob olevtyp e spaces

The principal goals in the section are the pro ofs of Theorems and We start with

an elementary lemma

Lemma Let and be nonnegative nite Radon measures on and let f g be a

sequence of bounded R valued Radon measures on If f g is uniformly ACand if

j j

denotes the LebesgueRadonNikodym decomposition of with respect to then fV g is

j j

uniformly summable

Proof This is easily seen by writing down the denitions

Lemma Let be a nonnegative nite Radon measure on and let fu g be a se

quence in BV R Assume that fDu g is uniformly AC and that fru g generates

j j

the Young measure Then thereexistsaL negligible set N such that for x n N

each is a probability measure with a centre of mass and Jensens inequality

x x

fd f

x x

holds for al l quasiconvex functions f R R for which f X jX j has a nite limit

as X

Proof The pro of pro ceeds in four steps

Claim without loss of generality we can assume that sup jDu j and that

n n

u u strongly in L R for some u BV R

Since mandjDu jA whenever H Aitfollows that fDu g is uniformly

j j

fxg Henceforth AC where is the nonatomic part of ie

x na na na

we shall supp ose that and the uniform AC then implies that sup jDu j

na j

Let B b e an op en ball contained in and dene

v xu x u y dy x B

j j j

Clearly Dv Du bB and by Poincares inequality it follows that fv g is b ounded in

j j j

BV B R By virtue of the RellichKondrachov Compactness Theorem there is a sub

sequence for convenience not relab elled and some v BV B R such that v v

n nm

strongly in L B R Note that frv g generates the Young measure bB R

dx Since it is sucient to prove that the Young measure has the claimed

x x

prop ertyonany ball contained in Step concluded

Consider the Leb esgueRadonNiko dym decomp ositions with resp ect to the Leb esgue

measure

m s m s

a L and Du ru L V

j j j

J Kristensen

Claim without loss of generality we can assume that u ru weakly in

nm nm

L R andV weakly in L R

m s

By Lemma fru g is uniformly L summable on and fV g is uniformly sum

j j

mable on Furthermore we deduce from Step that Du Du weakly in C and

using the uniform summability of the terms in the Leb esgue decomp ositions wededuce

ru ru weakly in L R

and

V V weakly in L R

The claim follows by considering the dierence fu ug using the observation made at

the end of Subsection and noting that the class of quasiconvex functions f for which

f X jX j has a nite limit as X is invariant under translation

Fix a quasiconvex function f R R with the prop erty f X jX j as

Claim there exists a negligible set N suchthatforx n N is a probability

f f

and measure

fd f

Let b e the set of all x in for which is a probability measure with The

x x

set has full measure in Fix x and take r such that B In view of

with X v and wehave

f ru dy f L B

x r

s s

jD u jD u ju y j dy

j j j

B nB

x r

Since ff ru g is uniformly summable and fru g generates the Young measure it

j j

follows from Theorem that

Z Z Z

f ru dy fd dy

j y

B B

xr xr

For the singular measure we notice that

s s

jD u jB jV j d

j xr j

s s

and since fjV jg is uniformly summable and is nonatomic there exist by the Dunford

Pettis Theorem a subsequence fjV jg and a g L such that jV j g weakly in

j j

k k

Passing to the limit through this subsequence we obtain L

Z Z Z

s m

fd dy gd f

L B

B B

xr xr

Lower semicontinuity

By Leb esgues Dierentiation Theorem the set of p oints x for which

m s

L B gd

and

Z Z Z

fd dy fd

y x

as r has full L measure If wedeneN n then for x n N we obtain

f f f

as r and then that

f fd

as claimed

Weshow that it is p ossible to nd a negligible set N which is indep endentof f

Let E denote the space of continuous functions g R R for which g X jX j has

a nite limit as X and dene the norm of g E as

jg X j

kg k sup

jX j

Hereby E is a separable Banach space and if therefore Q denotes the set of all quasiconvex

functions in E there exists a countable set F Q which is dense in Q We now dene

N and it is not dicult to showthat N has the desired prop erties N

f F

Proof of Theorem By Lemma there is a negligible set N such that for all x n N

each is a probability measure with a centre of mass and with the prop erty that Jensens

inequality

fd f

x x

holds for all quasiconvex functions f for which f X jX j has a nite limit as X

Since sup kru k it follows that for p

j p

Z Z

jX j d X dx

and hence that

jX j d X

for almost all x For p it follows that there exists a compact set K R such

that for almost all x

is supp orted in K

Let M denote the exceptional set in in case p and in in case p

The pro of is concluded since for x n N M itfollows from Lemma and Corollary

b is a gradient pYoung measure that L x

J Kristensen

Proof of Theorem As mentioned in the Intro duction it is p ossible to give a pro of

based on Theorem and Theorem We also mentioned that in the case p

it is p ossible to give a pro of which only uses Prop osition and Lemma We fo cus

on the latter leaving the details of the case p f g to the interested reader

We can assume that the lefthand side of is less than Because fF x u ru g

j j

is assumed uniformly summable it then follows that the lefthand side of is a real

number By extracting a subsequence if necessary we can assume that

F x u ru dx l R

j j

Put V u ru and consider the corresp onding sequence f g of elementary Young

j j j V

measures Since sup L fx jV xj tg as t there exists by Lemma

a subsequence for convenience not relab elled which generates a Young measure

Because u u lo cally in measure it follows that if

x x

then where

x x

is a Young measure generated by fru g In view of Theorem

Z Z

F x uxX d X dx l

As in the pro of of Lemma we see that ru ru weakly in L R and therefore

rux almost everywhere Let N be the negligible set from Lemma and put

fx n N ruxg We claim that for x the inequality

F x ux rux F x uxX d X

holds Together with this entails

Fix x and put f X F x uxX Then f is quasiconvex and by H and

Lemma

jf X j

lim sup

jX j

For an integer k dene f X jX j k maxff X k g Wenow apply Prop osition

to f and we apply Lemma to each of the approximating quasiconvex functions

Herebywe deduce that

Z Z Z

f rux f d jX j d X maxff k g d

x x x

k k

for all k Passing to the limit k the inequality follows

Lower semicontinuity

Lower semicontinuity in GSBV

Before pro ceeding with the pro of of Theorem we briey recall some denitions and

results from The following denition is motivated by some minimisation problems that

n n

are not co erciveinSBV R but in the enlargement GSBV R see Example

Denition L Ambrosio and E De Giorgi A Borel function u R is a

generalised special function of bounded variation briey u GSBV R if u

SBV for al l C R for which r has compact support

loc

Remarks Under the natural denitions of addition and multiplication with scalar

the space GS B V R isavector space

For b ounded functions there is nothing new as GSBV L SBV

loc

In this pap er we assume that n and then we have that a Borel function u

n n

GS B V R if and only if u SBV for all C R Recall that v

loc

SBV if and only if v j SBV for all op en subsets that are compactly contained

loc

Prop osition L Ambrosio Propositions and Let u GSBV R

Then u is approximately dierentiable almost everywhere in and the jump set S is

countably rectiable If N denotes a Borel measurable unit normal to S then it is

u u

possible to dene onesided traces u and u by the procedure described in Section

n n

Remark If u GS B V R and C R then

m m

D ururu L bu u N H bS

u u

Indeed by the Chain Rule for approximate derivatives r ux ruxrux

and since esp ecially is continuous u u onS S and u u

on S n S

The motivation for intro ducing the space GS B V R comes from the following com

pactness result To state it we need three test functions and g

Let beaconvex nondecreasing function satisfying the condition

as t

Let b e a concave nondecreasing function satisfying the condition

as t

Let g R b e a normal integrand satisfying the condition

g x v as v

for almost all x

J Kristensen

Theorem L Ambrosio Theorem Let and g be three functions as

described above Let fu g bea sequence in GS B V R and assume that

Z Z

sup g x u jru j dx ju u j dH

j j

Then there exists a subsequence fu g and a function u GS B V R such that u

j j

k k

u in measure on and ru ru weakly in L R Furthermore the function u

also satises the inequality

Remarks The last statement that u also satises is not explicit in However

it follows from the results in Section of that the integrand a b N ja bjis

regularly biconvex and then the pro of of Theorem in can b e used to show that

Z Z

m m

ju u j dH lim inf ju u j dH

j j

S S

u u

The lower semicontinuity of the bulk energy term follows from Fatous Lemma and the

convexity of X jX j

The function is subadditive This follows easily if we for xed s consider the

auxiliary function ht t s s t t and observe that it is nondecreasing

b ecause is concave

The main result of this section ensures by and Corollary lower semicontinuity

of quasiconvex integrals in the setting prescrib ed by this compactness result

Theorem Let fu g be a sequence in GSBV R which satises the boundedness

condition Assume that for some p sup kru k and that fru g

j p j

generates the Young measure

x x

Then for almost al l x the measure L b is a gradient pYoung measure

Remark We obtain Theorem in the sp ecial case where fu g SBV R and

g x v jv j

A brief outline of the pro of is as follows First it is shown utilising Theorem that

it is not restrictive to assume that u in measure and that ru weakly in L

j j

Next we truncate the functions u to obtain new functions v with the prop erties v

j j j

in L ru rv inL and by use of the VitaliHahnSaks Theorem Theorem

j j

below fDv g is uniformly AC for some We conclude using Theorem

Theorem Theorem p Let be a nonnegative nite Radon measure

on Assume that f g is a sequence of bounded R valued Radon measures on and

that each is AC If for al l measurable sets A the limit lim A exists in

j j j

then f g is uniformly AC R j

Lower semicontinuity

Proof of Theorem The pro of pro ceeds in three steps

It is not restrictive to assume that u in measure on and ru weakly in

j j

L R

By Theorem there exist a subsequence of fu g for convenience not relab elled

and u GS B V R such that u u in measure on and ru ru weakly

j j

nm n

in L R Dene v u u Then v GS B V R v in measure

j j j j

and rv weakly in L R By the remarks following Theorem and since

S S S

v u u

j j

Z Z

m m

j dH u j dH sup ju v sup jv

j j j j

j j

S S

u v

j j

This concludes Step

Let C b e such that tt if t and tif t Denote by

Lip its Lipschitz constant and notice that Lip Dene the radial mapping

if v jv j

jv j

if v

n n

For dene the rescaled function v v Clearly C R R and

Lip Lip v v if jv j and v if jv j

Dene u u

Claim there exist numbers suchthat fu gSBV R satises

ku k

kru ruk

and

Du A

for all Borel sets A

Clearly u SBV R

j loc

ku k sup j jsup jj

and

m m

Du ru L u u N H bS

u u

j j

By the Chain Rule for approximate derivatives ru r u ru where I denotes

j j

the n n unit matrix

u u ju j

j j j

I r u r u ju j ju j ju j

j j j j j

ju j ju j

j j

J Kristensen

In particular r u I when ju j and by Step we then deduce that

j j

ru ru in measure on

Since also jru j Lipjru j it follows that fru ru g is uniformly summable on

j j

and consequentlyby Vitalis Convergence Theorem

lim kru ru k

Put

M sup ju u j dH

j j

By M Since is continuous

S and S u u u u

u u

j j

and in particular ju u jc wherec supjj Next note that t inf f tt g

j j

has the same prop erties as b esides b eing at t Hence we can assume

that Because is concave the dierence quotient tt is nonincreasing hence

ju u j

j j

Lip

ju u jLip

j j

H almost everywhere on S Because is nondecreasing and Lip Lip

ju u j

j j

ju u j

j j

Lip

and together with this implies

sup ju u j dH LipM

j j

In view of it is p ossible to nd numbers suchthat

and kru ru k

j j

By and it follows that u SBV R that hold and

that for any Borel set A

Z Z

j dH u jDu Ajj ru dxj ju

j j

A AS

hence that holds

Dene for each j the nonnegative nite Radon measure

Z Z

j dH A B u ju A jru j dx

j j

A AS

j j

Lower semicontinuity

Dene for each h C the number

h hi

It is readily seen that hereby is a b ounded linear functional on C and that h hi

if h Therefore is a nonnegative nite Radon measure on

From the denition it follows that each Du is AC Let A be a measurable subset

of Since Radon measures are Borel regular there is a Borel subset B of such that

A B and B n A By AC and

Du ADu B

j j

and hence by the VitaliHahnSaks Theorem fDu g is uniformly AC The conclusion

nowfollows from Theorem

Proof of Theorem The pro of is analogous to the pro of for Theorem and is omitted

here Notice also that it is p ossible to state the result in terms of GSBV functions as

mentioned in the remark following Theorem

Examples and remarks

We discuss the various hyp otheses encountered in the pap er and give examples showing

that some are indisp ensable

Ad H The condition that F is a normal integrand do es not app ear to b e necessary

for Theorems and to hold Indeed in the case n much less is needed if the

integrand is autonomous as is shown in However for the metho d employed in this

pap er it seems that being a normal integrand is close to the weakest p ossible regularity

assumption on F

Under the assumption that F is a nonnegative Caratheo dory integrand satisfying the

growth condition H the result of states that quasiconvexityofF x v X inX for

almost all x and all v is necessary and sucient for I u to be sequentially weakly lower

semicontinuous on W and similarly for p So far no necessary condition has b een

found if F is merely assumed to b e a normal integrand The condition should b e related

to quasiconvexity but it is likely that it also involves the v variable

Ad H Quasiconvexity is the natural assumption in the multidimensional case It is

however very hard to verify that a given function is quasiconvex and partly for this reason

rank convexity and p olyconvexity have b een studied in the calculus of variations The

nm nm

function f R R fg is rank convex if it is convex on rank lines in R

and it is p olyconvex if f X is a convex function of the minors of X eg if m n f

is p olyconvex if f X hX det X where h is convex

If f is C then rank convexityoff is equivalent to the LegendreHadamard condition

for r f r f X a b a b X a b The notion of p olyconvexitywas intro duced

by Ball in some sp ecial cases app ear implicit in and is related to null

J Kristensen

Lagrangians and to weak sequential continuity see also and For realvalued

functions wehaveschematically

p olyconvex quasiconvex rank convex

If minfm ng then the concepts reduce to ordinary convexity whereas for m n

there are quadratic p olyconvex functions that are not convex For m n there exist

quasiconvex functions that are not p olyconvex see and the references therein Rank

convexity is equivalent to quasiconvexity for quadratic forms in all dimensions and if

minfm ng then it is even equivalent to p olyconvexity see and the references

therein It is not dicult to show that the same is true for p olynomials of at most third

degree However as shown by Sverak in there are p olynomials of degree on R

which are rank convex but not quasiconvex when n m This example conrmed

a conjecture from for dimensions n m Morreys conjecture that rank

convexity do es not imply quasiconvexity is still op en in dimensions n m We

refer to for a further discussion of the matter

Ad H The lower semicontinuity results fail without the growth condition H

p p

cf In Ball and Murat observed that in the case where F x v X f X

is bounded from below but allowed to be a necessary condition for I u to be se

quentially weakly lower semicontinuous on W is that the condition holds for all

u W R They called this strengthening of the ordinary quasiconvexity condition

for W quasiconvexity ordinary quasiconvexity corresp onds to W quasiconvexity

It is not hard to see that if F x v X f X satises H then quasiconvexity is equi

p p

valentto W quasiconvexity The W quasiconvexity condition dep ends in a dramatic

wayonp As a consequence of Theorem it follows that for m n the function

f X jdet X j is W quasiconvex if and only if p m see also It is still an op en

question whether W quasiconvexity together with some regularityoff eg continuity

is sucient for sequential weak lower semicontinuityof I on W to o Wenoticethatby

Example of there are lower semicontinuous functions f whichare W quasicon

vex but not rank convex A result of Tartar p states also in this generality

that rank convexity is a necessary condition for sequential weak lower semicontinuityof

I Hence some additional assumption is needed in general for W quasiconvexitytobea

sucient condition for sequential weak lower semicontinuityonW See Theorem

and Conjecture and the remarks afterwards Partial results have b een obtained in

and

In Pedregal observed that if the W quasiconvexity condition is slightly strengthened

it b ecomes sucient Following Pedregal a Borel function f R R fgwhichis

p nm

b ounded from b elow is closed W quasiconvex if for all probability measures on R

for whichL b is a gradient pYoung measure Jensens inequality holds for f and

Z Z

fd f where XdX

Whether this condition is necessary for I to be sequentially weakly lower semicontinu

ous on W to o is still an op en problem Note that under the growth condition H

quasiconvexity is equivalent to closed W quasiconvexity and that the results in Theor

ems and remain valid if instead of H and H we assume that F x v X is p

Lower semicontinuity

closed W quasiconvex in X Notice also that with this assumption F is allowed

An example in shows that the structure of gradient pYoung measures can b e rather

complicated Consequently the question of whether for suciently regular functions

p p

W quasiconvexity is equivalent to closed W quasiconvexity might b e subtle Partial

results in this direction have b een obtained in

The next result shows the signicance of H for multiple integrals I with integrands

F x v X f X The result is wellknown and is a sp ecial case of a general result due

to Alb erti Our pro of is elementary and cannot be adapted to treat the general case

considered in

Let f R R fg be a lower semicontinuous function which is b ounded from

p n

b elow Let p u W R and dene

p n n

A fu W R u u W R g

I u f ru dx

Prop osition The functional I is nite on A if and only if there exists a constant

c such that

nm p

X R f X c jX j

Remark Instead of requiring I on A it is enough if fu A I u g has an

p n

interior p ointinAW R

p n

Proof The if part is trivial Assume that I on A and consider AW R

as a complete metric space Without loss in generality we can assume that f By

Fatous Lemma I is lower semicontinuous on A Since

fu A I u tg A

it follows from Baires Theorem that for some t the sublevel set fu A I u tg has

nonemptyinterior Assume that for u A and

I u t

holds whenever u A and ku u k

We next construct a test function u which together with yields the conclusion

Dene as s if s and s s if s Take

x tobeapLeb esgue p ointof ru and consider r distx For X R

dene

x x

m r

X x x u xx R v x

and

r r

u v u A

J Kristensen

Notice that for some c c m n p

r r mp

ku u k kv k c jX jr kru k

p p pB

x r

for r distx

Fix c c jru x j and take R distx such that

r mp

ku u k c jX jr

if r R Clearly

r r m

I u f ru dx L B f X

x r

and hence taking

r min R

c jX j

we infer from that

m m

t L B min R f X

c jX j

f X c jX j

for some suitable c which is indep endentof X

Recall that any convex function f R R is the p ointwise limit of an increasing

sequence of convex functions of linear growth at innity The results in imply that

a similar statement is false for quasiconvex functions The next example shows that the

situation do es not improveifwe relax the requirement and only try to approximate with

rank convex functions Before turning to the details of this we rst observe a simple

consequence of a result due to Sivaloganathan see also

m m

Let andB fx R jxj g A function u B R is called radial if

there exists a function R R such that for almost all x

R jxj

x ux

jxj

p m

Recall see eg that for m and p the radial function u b elongs to W B R

if and only if R R can b e taken absolutely continuous and such that

R r

m p

r jR r j dr

Prop osition Let f R R be a rank convex function which for some p

satises the growth condition

jf X j

lim sup

jX j

Lower semicontinuity

mm m

Then for X R and radial functions u W B R the inequality

f X ru dx f X

holds where B denotes the open unit bal l in R

Proof We can assume that X By the inequality holds if u is a smo oth radial

function Fix any radial function u in W B R and take Using a standard

mollier argumentwe can nd smo oth radial functions u B R where u x

j j

if jxj suchthat ku u k where we have extended u by outside B By

j pB

the growth assumption

Z Z Z

f ru dx f ru dx f ru dx f L B n B

B B B

and since u is a smo oth radial function vanishing at B

f ru dx f L B

we deduce

Prop osition Let f R R fg be a function with the property that it can

be approximated from below by rank convex functions f ief f and f f point

j j j

wise each f verifying the growth condition for some xed p Then holds

for f and al l radial functions u W B R

Proof This is easy

Example Let p and dene f X jX j jdet X j X R Then f is

p olyconvex and satises the p growth condition

jX j f X jX j

We claim that f cannot b e approximated from b elow with rank convex functions f that

grows p olynomially slower at than f ie for some q

jf X j

lim sup

jX j

holds for each j The following argument is inspired by the pro of of Theorem in

Reductio ad absurdum assume that for some q this is p ossible Then by Prop osition

the inequality holds for f and all radial functions u W B R However

this cannot b e true Indeed for intro duce the radial function

jxj

u x x x B

jxj

J Kristensen

Observe that u W B R and therefore by the assumption and Prop osition

I and u u in taking X

p p

f I ru dx f I

p p

I and Since f

f I ru dx c

we obtain

p p p

Since p this is a contradiction if is large enough

Ad and That it is necessary to imp ose some additional condition on the

sequence of negative parts follows from Counterexample in There it is shown that

if m n and F x v X detX thenI u is not sequentially weakly lower semicon

tinuous on W In Meyers found a technical condition which might be relevant in

this connection However no attempt has b een made to investigate this condition in the

present setting

The content of the next example is wellknown and shows on the level of Young measures

the dierence b etween the multidimensional case m n considered in this pap er and

the scalar case minfm ng

nm th

Example Let p and be a Young measure on R with a nite p

order moment Assume that

ae dx and

x x x

If m or n then is a gradient pYoung measure For m n this is in general

false

The rst assertion follows b ecause quasiconvexityisjustconvexity in the case minfm ng

The second assertion follows from the fact that if m n there are quasiconvex func

tions which are not convex If p we can takeany second order minor and if p

the examples are provided in eg

The following example concerns the hyp otheses and in Theorems and

nm th

Example Let p and be a Young measure on R with a nite p

order moment Assume that

dx and ae

x x x

Lower semicontinuity

There exists a sequence fu gSBV R with the prop erties

ku k sup kru k ju u j dH

j j p

j j

fjru j g is uniformly summable on if p

and

weakly in C R

Proof For simplicity we assume that B fx R jxj g and that is

homogeneous L bB We also assume that p the case p can be

treated analogously

p nm p

There is a sequence fV gsuch that V weakly in L B R fjV j g is uniformly

j j j

summable on B and fV g generates see It is not restrictive to assume that each

V is compactly supp orted in B In view of Theorem in applied to eachrowofV we

j j

can nd v SBV B R with the prop erties

rv V and jv v j dH ckV k

j j j B

j j

where c is a constant dep ending on m and n only Because V is compactly supp orted in

B we can also take v to b e compactly supp orted in B

Put t kv k and consider the family F of all closed balls of radius less than

j j B j

jt which are contained in B By Vitalis Covering Theorem there exists an at most

countable subfamily F whichcovers L almost all of B and consists of pairwise disjoint

balls Write F fB g and dene

j j

k K

x r

k k

x x

u xr v if x B k K

j j

j j j

x r

k k

Hereby u is welldened and it is readily veried that u SBV B R that

j j

ku k j jDu jB jDv jB ckV k

j B j j j B

p nm

and that fjru j g is uniformly summable on B Fix C B and C R and

compute

Z Z

j j j

r x r x V x dx ru dx

j j

k k k

B B

k K

Since is uniformly continuous

j j j

r x r x dy

k k k

k K

uniformly in x B and consequently

Z Z Z

ru dx dx d

B B

The next example concerns the hyp othesis in Theorems and

J Kristensen

Example Let be a Young measure on R with a nite rst order moment

and assume that

ae dx and

x x x

There exists a sequence fu gC R such that

ku k sup kru k

j j

and

weakly in C R

Proof As in Example but instead of applying Theorem of we apply Theorem

with p and j We leave the details of this to the interested reader

As mentioned in the Intro duction there are sequences fu g of functions satisfying the

conditions of Theorems but that do not allowa decomp osition ru rv E

j j j

where fE g converges strongly in L and fv g converges weakly in W Before giving

j j

the example we state an auxiliary lemma

It is well known that given anyvector eld V of class C on R there exists a function

whose gradient is V if and only if curlV where curlV is the function of R into

R dened by

V V

s r

curlV for rs m

x x

r s

By a mollier argument a similar result may be proved when V is a distribution and

curlV in the distributional sense

m m

Lemma Let fV g be a sequence converging weakly to V in L R R If V

j j

rv E fE g converges strongly in L and fv g converges weakly in W then curlV

j j j j j

curlV strongly in W for al l p

loc

Here we recall that h strongly in W means that for each op en and b ounded

loc

subset R and p pp wehave

hh i

uniformly in W withkk

Proof Supp ose that v v weakly in W and E E strongly in L Because

j j

V rv E and

V V rv v E E

j j j

we can without loss in generality assume that V E and v

Let tM supkV k Then wehave that

Z Z

jE j dx jrv j dx

j j

t M

fjE jtM g fjrv jtg

j j

Lower semicontinuity

By Lemma we may nd a sequence fw g which is b ounded in W R and such

that v w strongly in W R Since E rv rw V rw weakly

j j j j j j j

for all p and in L and strongly in L wehave that V rw strongly in L

j j

loc

thus the claim of the lemma follows

Example Let B denote the op en unit ball in R and dene for u W B the

functions u SBV R as

u x if jxj

u x

if jxj

If u weakly in W B then u strongly in L R

j j

m m

H S H BandfD u g is uniformly AC

u j

m m

where L bB H bB However in general fcurlru g do es not converge strongly

to in W for p Indeed assume for simplicity that m and let C R

loc

Then

s s

ru D u D u ru

j j j j

i h i dH u h

x y x y x y

thus if u on B then for each p wehave that

sup jh curlru ij

where the supremum is taken over say C B with kk

App endix

This app endix contains a pro of of the characterisation of gradientYoung measures The

general result we set out to prove is the following where we note that it diers from the

results of Kinderlehrer and Pedregal only in c where we test with rather sp ecial

quasiconvex functions

Theorem Let p and let dx be a Young measure Then is a

x x

gradient pYoung measure if the fol lowing three conditions are satised

a has a nite p order moment

rux almost everywhere b there exists u W R such that

c for al l quasiconvex functions f R R satisfying f X f X jX j for

jX j large the Jensen inequality

fd f

x x

holds for almost al l x

J Kristensen

Conversely if is a gradient pYoung measure then thereexistsaL negligible set N

such that for any quasiconvex function f R R verifying the growth condition

f X

if p lim sup

jX j

and no growth condition if p then the Jensen inequality

fd f

x x

holds for al l x n N

Some of the technical ingredients in the pro of are taken from however the pro of

relies on the HahnBanach Separation Theorem and is in this sense similar in spirit to the

original pro ofs in The key p oints distinguishing our pro of are the use of Corollary

or Prop osition the choice of function space we treat the case of inhomogeneous

Young measures directly and the observation contained in Lemma

We start with a lemma which in the case p is identical to Theorem in The

general result can be inferred from the results in but we give a selfcontained pro of

essentially following the strategy prop osed in for the case p

Lemma Let p and B fx R jxj g If is a gradient pYoung

measure and dx then there is a L negligible set N such that for

x x

x n N the measure L bB is a gradient pYoung measure

Proof We only give the pro of for the case p The remaining cases can be

treated analogously By assumption there exists a sequence fu g such that for some

p n

u W R

p n

u u weakly in W R

and

weakly in C R

Extracting a subsequence if necessary we can assume that

p m

jru j L b weakly in C

where is a nonnegative nite Radon measure on the closure of Let f g and f g

k l

be two sequences which are dense in C B andC R resp ectively We can assume

that each C B Extend by outside B and put

k k

x d

In view of Theorem wehave for x andr distx

y x

ru y dy lim r x

l kr k

Lower semicontinuity

where y r yr and denotes convolution Let denote the set of p oints

kr k

x where

x dy x lim

k kr

l l

x R lim

m m

L B dL

and

p p

lim r juy ux ruxy xj dy

Put N n and notice that L N Fix x n N and intro duce for r

distx the rescaled functions u and u as resp ectively

u x ry uxy B u y

and

ux ry uxy B u y

For convenience of notation put

Z Z

x I j r y ru y dy dy

k l k

l k i

Z Z

xr xr

p p

ju y ruxy j dy sup jru y j dy x

j j

B B

and notice that lim lim I j r for each i For each i take rst r i

j i i

suchthat

lim I j r i

i i

p n

and notice that fv g W B R and next j isoI j r i Dene v u

i i i i i i

has the prop erties

p n

v rux weakly in W B R

and

Z Z Z

lim rv dy dy d

i x

k l k l

B B

for all k l The pro of is concluded by noticing that the linear span of tensor

B R pro ducts is dense in C

k l

Proof of the second part of Theorem Apply Lemma and Corollary

In the remainder of this section we fo cus on proving the rst part of Theorem In

view of Corollary it suces to show that is a gradientYoung measure Since is

a gradientYoung measure if and only if dx is and we

x x x

rux

can assume that u Hence by Lemma we can assume that the measure

x x

satises the conditions

J Kristensen

jX j d x X a

b for almost all x

c for all quasiconvex functions f R R satisfying f X jX j as X

the inequality

fd f

holds for almost all x

We intro duce some notation for the pro of The space of Lipschitz functions F

R R is a nonseparable Banach space with the norm

kF k jF x j LipF

where x is arbitrary but xed and LipF denotes the Lipschitz constantofF The

Lipschitz constant refers to the metric distx X y Y jx y j jX Y j Let E

denote the subspace of Lipschitz functions F R R with the prop ertythatthe

function

F x X

x X

jX j

admits a continuous extension to R fg the closure of times the onep oint

compactication of R with the natural metric top ology It is readily veried that E

is a closed subspace and hence that E kk is a Banach space E is nonseparable but

that is immaterial for our purp oses

Let P denote the set of p ositive measures on R satisfying

nm m

A R L A for all Borel sets A and jX j dx X

Let Y denote the subset of P consisting of measures with the additional prop ertythat

R suchthat there exists a sequence fu g in W

u weakly in W R

and

weakly in C R

We regard P as a subset of the dual space E by the duality pairing

h F i FdF E

and herebywehave that

kk sup h F ic j j d

kF k

where c is a constant dep ending on the diameter of only

We are going to showthat Y by using the HahnBanach Separation Theorem in the

dual space of E Before pro ceeding to the details of this we need some auxiliary results

Lower semicontinuity

Y denotes the weak closure of Y in E then Y P Y Lemma If

Proof Fix Y P Take f g W which is dense in C and ff g

i j

nm nm nm

W R which is dense in C R Dene F x X jX j x X R

and notice that F E Since also f belongs to E for all i j it follows that for

i j

each p ositiveinteger k there exists in Y such that whenever i j k

jh F ij jh f ij

i j

k k

By denition of Y and in view of Theorem we maytake u W R verifying

for i j k

Z Z Z Z Z

ju j dx j jru j dx jX j d x X j j f ru dx f d j

i j i j

k k k k k

Thus wehave in particular for i j k

Z Z Z Z

f d j j jru j dx jX j d x X j j f ru dx

i j i j

k k

and therefore

Z Z

lim f ru dx f d

i j i j

for all integers i j and

Z Z

lim jru j dx jX j d x X

We infer that

Z Z

lim f ru dx fd

for all f C C R and hence that fru g generates the Young measure

By and Theorem fru g is uniformly summable and therefore fu g converges

k k

weakly to in W R This proves that Y

The next result is an approximation result It states that a general Young measure in

Y can b e approximated by piecewise constantYoung measures from Y In the statement

of the lemma we denote by G the collection of all op en dyadic cub es Q of sidelength

contained in that is

k m

G fQ x x has integer co ordinatesg

Lemma Let dx Y and let k beapositive integer If we dene P

x x

Z Z Z

h f i x fd dy dx

Q Q

for C then Y and weakly in E f C R

k k

J Kristensen

almost everywhere in we may apply the averaging principle see Proof Since

eg Theorem in or to deduce that for each Q G the probability measure

dened as

Z Z

h f i fd dy f C R

Q y

gives rise to a homogeneous gradientYoung measure L bQ Dene R n G

k k

m m

and note that L bR L bQ it follows from this that Y

k k k

Considered as a linear functional on E the norm of is

Z Z

k kc j j d dx

thus the sequence f g is normb ounded in E To conclude the pro of wex W

f W R suchthat f E and compute

Z Z Z

h f i x fd dy dx

Q Q

Because the function x fd is summable it follows that h f ih f i and

the lemma follows from this

Lemma If co Y denotes the weakly closedconvexhullofY in E then coY P Y

Proof By Lemma it suces to prove that Y is convex In view of Lemmata and

m m

it is enough to showthatif L b and L b are two homogeneous

measures in Y then also t t belongs to Y for each t

To start the pro of we make the following observations Let U be a nonempty op en

m n

b ounded subset of R letu W U R and dene the probability measure as

h f i f ru dx f C R

and L b Using the generalised RiemannLeb esgue lemma see eg

Theorem p it is easy to show that b elongs to Y and by Lemma it is

clear that any homogeneous measure in Y is the weak limit in E of some sequence f g

where u b elong to W R

Take op en and b ounded subsets U U R satisfying

m m

U U and L U tL U U

Take fu g W U R suchthat

ij i

F i h F iF E i lim h

i u

U U R as Dene v W

u x if x U

v x

u x if x U

Lower semicontinuity

Clearly Y and for F E

lim h F i ht t F i

Y The claim follows from Lemma hence t t

Proof of rst part of Theorem In view of Lemma and since P it suces to

show that co Y In order to prove this wenoticethatby the HahnBanach Separation

Theorem

co Y fH H weakly closed halfspace containing Y g

Let H be a weakly closed halfspace in E that contains Y By denition there exists

a weakly continuous linear functional T E R and a number t R such that

H fl E T l tg A weakly continuous linear functional is an evaluation

functional ie T l hl Gi l E for some G E cf and since Y H

h Git

for all Y We claim that this implies that

qc qc

G E and G x dx t

where for each x G x denotes the quasiconvex envelop e of Gx

By hyp othesis c this entails that b elongs to H Hence we conclude the pro of by

verifying

Recall that for a p ositive integer k G denotes the family of all op en dyadic cub es of

m n k

R and extend sidelength contained in For each Q G pick u W

m m n

each u to all of R by periodicity Dene v W R as

Q j

u j x if x Q Q G

v x

otherwise

Then v weakly in W R andfrv g generates the Young measure

j j x

dx where by use of the notation from the pro of of Lemma wehave

if x Q Q G

otherwise

Clearly Y and so

Z Z Z

t Gx dx Gx ru y dy dx

n G Q

For each Q G we let x bethe lower left corner p ointof Q Since G is Lipschitz

continuous and the diameter of each Q is m weget

Z Z

k m m

t m LipGL Gx dx L Q Gx ru y dy

Q Q

n G

QG k

J Kristensen

Gx ru y dy t L Q

Q Q

where is indep endent of the functions u and tends to as k tends to Taking

m n

the inmum over u W R yields the inequality

m qc

t L QG x

which is valid for each k By Lemma b elow it follows that G b elongs to E and

therefore we conclude the pro of by letting k tend to innityand noticing that the right

hand side is a Riemann sum for the integral G x dx

Lemma Let F E and assume that there exists x X R such that

qc qc

F x X Then F E

Proof For x X andy Y wehave

F x X F y Y LipF jx y j jX Y j

Taking y x and Y X yields

qc qc

F x X F x X LipF jx x j

and therefore it follows that F is realvalued for all x X Next we take Y X H

and get

qc qc

F x X F y X H LipF jx y j jH j

qc qc

wherebywe conclude that F is Lipschitz continuous It follows easily that F x X jX j

has a nite limit as X and that this limit is indep endentof x

Acknowledgements

It is a pleasure for me to record my thanks to G Alb erti S Demoulini S Muller and

PPlechac for stimulating discussions related to the sub ject of this pap er Parts of the work

was carried out while the author was visiting the MaxPlanck Institute for Mathematics

in the Sciences Leipzig The supp ort from the Danish Natural Science Research Council

and from MPI is gratefully acknowledged

References

E Acerbi and N Fusco Semicontinuity problems in the calculus of variations

Arch Rat Mech Anal

G Alb erti A Luzin typ e prop erty of gradients J Funct Anal

G Alb erti Rank one prop erties for derivatives of functions with b ounded variation

Proc Roy Soc Edinburgh A

Lower semicontinuity

G Alb erti Integral representation of lo cal functionals Ann Mat pura ed Appl

IVCLXV

JJ Alib ert and G Bouchitte Nonuniform integrabilityand generalized Young meas

ures J Convex Anal

L Ambrosio Existence theory for a new class of variational problems

Arch Rat Mech Anal

L Ambrosio On the lower semicontinuity of quasiconvex integrals in SBV Nonlinear

Analysis

L Ambrosio G Buttazzo and I Fonseca Lower semicontinuity problems in Sob olev

spaces with resp ect to a measure J Math Pures Appl

L Ambrosio and G Dal Maso On the relaxation in BV R of quasiconvex integrals

J Funct Anal

L Ambrosio and E De Giorgi Un nuovo tip o di funzionale del Calcolo delle Variazioni

Atti Accad Naz Lincei Cl Sci Fis Mat Nat

EJ Balder A general approach to lower semicontinuity and lower closure in optimal

control theory SIAM J Control and Optimization

JM Ball Convexity conditions and existence theorems in nonlinear elasticity

Arch Rat Mech Anal

JM Ball A version of the fundamental theorem for Young measures In PDEs and

Continuum Models of Phase Transitions Eds M Rascle D Serre M Slemro d Springer

Verlag Lecture Notes in Physics

JM Ball JC Currie and PJ Olver Null Lagrangians weak continuity and variational

problems of arbitrary order J Funct Anal

JM Ball and F Murat W quasiconvexity and variational problems for multiple

integrals J Funct Anal

JM Ball and F Murat Remarks on rankone convexity and quasiconvexity In

Proc Dundee Conference on Di Eq

JM Ball and K Zhang Lower semicontinuity of multiple integrals and the Biting

Lemma Proc Roy Soc Edinburgh A

H Berlio cchi and JM LasryIntegrandes normales et mesures parametrees en calcul

des variations Bul l soc math France

G Bouchitte I Fonseca and L Mascarenhas A global metho d for relaxation Preprint

N Bourbaki Integration Chap IX Elements de Mathematique Hermann Paris

B Dacorogna Direct Methods in the Calculus of Variations SpringerVerlag

N Dunford and JT Schwartz Linear operatorsVol Interscience

LC Evans and RF Gariepy MeasureTheory and Fine Properties of Functions Studies

in advanced mathematics CRCPRESS

I Fonseca and S Muller Relaxation of Quasiconvex Functionals in BV R for In

tegrands f x u ru Arch Rat Mech Anal

I Fonseca and SMuller Aquasiconvexity lower semicontinuity and Young measures

Preprint

J Kristensen

I Fonseca S Muller and P Pedregal Analysis of concentration and oscillation eects

generated by gradients Preprint

E De Giorgi G Buttazzo and G Dal Maso On the lower semicontinuity of certain

integral functionals Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend

P Ha jlasz A note on weak approximation of minors Ann IHP Analyse Non Lineaire

N Hungerbuhler A renement of Balls theorem on Young measures COCV

T Iwaniec and C Sb ordone Weak minima of variational integrals J reine angew Math

A Kalama jska On lower semicontinuityofmultiple integrals Col loq Math

D Kinderlehrer and P Pedregal Characterizations of Young measures generated by

gradients Arch Rat Mech Anal

D Kinderlehrer and P Pedregal Gradient Young measures generated by sequences in

Sob olev spaces J Geometric Anal

J Kristensen Finite functionals and Young measures generated by gradients of Sob olev

functions Math Inst Technical University of Denmark MATRep ort No

J Kristensen A necessary and sucient condition for lower semicontinuity Preprint

M Kruzik and T Roubicek Explicit characterization of L Young measures

J Math Anal Appl

FC Liu A Luzin typ e prop erty of Sob olev functions Indiana Univ Math J

J Maly Lower semicontinuity of quasiconvex integrals Manuscripta Math

J Maly Weak lower semicontinuity of p olyconvex integrals Proc Royal Soc Edinb

P Marcellini Approximation of quasiconvex functions and lower semicontinuityofmul

tiple integrals Manuscripta Math

P Marcellini On the denition and the lower semicontinuity of certain quasiconvex

integrals Ann I H P Analyse Non Lineaire

NG Meyers Quasiconvexity and the semicontinuityofmultiple variational integrals of

any order Trans Amer Math Soc

CB Morrey Quasiconvexity and the semicontinuity of multiple integrals Pacic

J Math

CB Morrey Multiple integrals in the Calculus of Variations SpringerVerlag

S Muller Variational mo dels for microstructure and phase transitions Lectures at the

CIME summer scho ol Calculus of variations and geometric evolution problems

Cetraro

S Muller A sharp version of Zhangs theorem on truncating sequences of gradients Max

PlanckInstitut fur Mathematik in den Naturwissenschaften Leipzig preprint no

Lower semicontinuity

P Pedregal Jensens inequality in the calculus of variations Dierential and Integral

Equations

PPedregal ParametrizedMeasures and Variational Principles PNLDE Birkhauser

YG Reshetnyak General theorems on semicontinuity and on convergence with a func

tional Siberian Math J

T Roubicek Relaxation in Optimization Theory and Variational Calculus W de

Gruyter

W Rudin Real and Complex Analysis McGrawHill

W Rudin Functional AnalysisTata McGrawHill New Delhi th Reprint

J Serrin A new denition of the integral for nonparametric problems in the calculus of

variations Acta Math

J Sivaloganathan Implications of rank one convexity Ann IHP Analyse Non Lineaire

V Sverak Rankone convexity do es not imply quasiconvexity Proc Royal Soc Edin

burgh A

M Sychev Young measure approachtocharacterization of b ehaviour of integral func

tionals on weakly convergent sequences by means of their integrands Preprint

L Tartar Comp ensated compactness and applications to partial dierential equations In

Nonlinear Analysis and Mechanics HeriotWatt Symp osium IV Ed Knops RJ Pitman

Res Notes in Math

B Yan On the L mean co ercivity and relaxation of multiple integral functionals Pre

LC Young Generalized curves and the existence of an attained absolute minimum in the

calculus of variations Comptes RendusdelaSoc des Sciences et de Lettres de Varsovie

classe III

K Zhang A construction of quasiconvex functions with linear growth at innity

Ann Sc Norm Sup Pisa Serie IV XIX

WP Ziemer Weakly Dierentiable Functions GTM SpringerVerlag