Session 38. Variational Methods in Nonlinear Analysis

On one variant of Decomposition Lemma dealing with weakly converging sequences of gradients and applications to nonconvex variational problems

Agnieszka Kałamajska, Institute of Mathematics, University of Warsaw, Poland
We obtain the variant of Decomposition Lemma due to Kinderlehrer and Pedregal asserting that an arbitrary bounded sequence of gradients of Sobolev mappings \(\{ \nabla u_k\}\subseteq L^p(\Omega, {\bf R}^{m\times n})\), where \(p>1\), can be decomposed into a sum of two sequences of gradients of Sobolev mappings: \(\{ \nabla z_k\}\) and \(\{\nabla w_k\}\), where \(\{ \nabla z_k\}\) is equintegrable and carries the same oscillations, while \(\{ \nabla w_k\}\) carries the same concentrations as \(\{ \nabla u_k\}\). In our setting we additionally impose the general trace condition ``\(u_k=u\)'' on \(F\), where \(F\) is given closed subset of \(\bar{\Omega}\) and we show that under this assumption the sequence \(\{ z_k\}\) in decomposition can be chosen to satisfy also the trace condition \(z_k=u\) a.e. on \(F\).

The result is applied to nonconvex variational problems to regularity results for sequences minimizing functionals. As the main tool we use DiPerna Majda measures.

References
  1. I. Fonseca, S. Müller, P. Pedregal, Analysis of concentration and oscillation effects generated by gradients,, SIAM J. Math. Anal. 29 (1998), pp. 736-756.
  2. A. Kałamajska, On one method of improving weakly converging sequence of gradients , Asymptotic Analysis 62 (2009), 107-123.
  3. 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, No. 2 (2013), 545-571.
  4. D. Kinderlehrer, P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces J. Geom. Anal. 4 (1994) pp 59-90.
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