Abstract

We deal with the variant of Decomposition Lemma due to Kinderlehrer and Pedregal asserting that an arbitrary bounded sequence of gradients of Sobolev mappings {∇uk}⊆Lp(Ω,Rm×n)\{\nabla u_k\} \subseteq L^p(\Omega,\rmn), where p>1p>1, can be decomposed into a sum of two sequences of gradients of Sobolev mappings: {∇zk}\{\nabla z_k\} and {∇wk}\{\nabla w_k\}, where {∇zk}\{\nabla z_k\} is equintegrable and carries the same oscillations, while {∇wk}\{\nabla w_k\} carries the same concentrations as {∇uk}\{\nabla u_k\}. We additionally impose the general trace condition ``uk=uu_k=u'' on FF, where FF is given closed subset of Ωˉ\bar{\Omega}. We show that under this assumption the sequence {zk}\{z_k\} in decomposition can be chosen to satisfy also the trace condition zk=uz_k=u a.e. on FF. The result is applied to nonconvex variational problems to regularity results for sequences minimizing functionals. As the main tool we use DiPerna Majda measures.

Author information

Contact details are reproduced from the original publication and may be historical.

Agnieszka Kałamajska

Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02–097 Warszawa, Poland

kalamajs@mimuw.edu.pl

Keywords

  • Sequences of gradients
  • DiPerna Majda measures
  • concentrations
  • oscillations

Mathematics Subject Classification

46E35, 49J45, 35B05

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Suggested citation

A. Kałamajska. “On one Extension of Decomposition Lemma Dealing with Weakly Converging Sequences of Gradients with Application to Nonconvex Variational Problems.” Journal of Convex Analysis 20 (2013), No. 2, 545–571. https://doi.org/10.68381/jca20033

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