On one Extension of Decomposition Lemma Dealing with Weakly Converging Sequences of Gradients with Application to Nonconvex Variational Problems
Agnieszka Kałamajska · Journal of convex analysis · 2013
We deal with the variant of Decomposition Lemma due to Kinderlehrer and Pedregal asserting that an arbitrary bounded sequence of gradients of Sobolev mappings \{ abla u_k\} \subseteq L^p(\Omega,{{\bf R}}^{m\times n}) { ∇ u k } ⊆ L p ( Ω , R m × n ) , where p>1 p > 1 , can be decomposed into a sum of two sequences of gradients of Sobolev mappings: \{ abla z_k\} { ∇ z k } and \{ abla w_k\} { ∇ w k } , where \{ abla z_k\} { ∇ z k } is equintegrable and carries the same oscillations, while \{ abla w_k\} { ∇ w k } carries the same concentrations as \{ abla u_k\} { ∇ u k } . We additionally impose the general trace condition “ u_k=u u k = u ” on F F , where F F is given closed subset of \bar{\Omega} Ω ˉ . We show that under this assumption the sequence \{z_k\} { z k</