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Lower Semicontinuity in Spaces of Weakly Differentiable Functions

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Lower Semicontinuity in Spaces of Weakly Differentiable Functions MaxPlanckInstitut fur Mathematik in den Naturwissenschaften Leipzig Lower semicontinuity in spaces of weakly dierentiable functions by Jan Kristensen PreprintNr Lower semicontinuity in spaces of weakly dierentiable functions y 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 n 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 Z I u F x u ru dx m 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 n 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 m 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 y 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 d 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 t as t t n and if fu gW R satises j Z Z dD u j sup ju j dx d j d j n then for some subsequence of fu g for convenience not relab elled and some u W R j dD u dD u j weakly in L u u strongly in L and j 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 p 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 p F x v X if p lim sup p jX j X and no condition is required if p It is not dicult to show see Lemma that under the hyp othesis H the condition p H is equivalent to the condition lim sup jF x v X jjX j p p X Dening F inf fF gwehave the following result Lower semicontinuity Theorem Let be a nonnegative and nite Radon measure on p and n 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 j then Z Z F x u ru dx lim inf F x u ru dx j j j Remark It follows from the pro of that if we replace the assumption with dD u j sup p d j L then wehave instead of the conclusion Z Z ru ru j adx lim inf F x u adx F x u j k a a m 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 s ru dD u s E u adx d F x u Gx s a d Then the principal result of guarantees lower semicontinuityof E u on sequences fu g j satisfying and when F is a Caratheo dory integrand satisfying H and H p 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 m 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 u 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 u n 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 n 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 t as t t n and if fu gSBV R satises j Z Z m sup ju j jru j dx ju u j dH j j j j j S u j n then for some subsequence of fu g for convenience not relab elled and some u GS B V R j u u in measure ru ru weakly in L j j Z m j dH u ju and sup j j j S u For convenience we state a precise version of the compactness theorem for GSBV in Section n 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 j then Z Z F x u ru dx lim inf F x u ru dx j 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 t Wecover exactly the cases describ ed in the compactness theorem for GSBV Consider the functional Z Z m E u F x u ru dx Gx u u N dH u S u 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 n 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
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