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Session 38. Variational Methods in Nonlinear Analysis
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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
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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.
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References- I. Fonseca, S. Müller, P. Pedregal,
Analysis of concentration and oscillation effects generated by gradients,, SIAM J. Math. Anal. 29 (1998), pp. 736-756.
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A. Kałamajska, On one method of improving weakly converging
sequence of gradients , Asymptotic Analysis 62 (2009), 107-123.
- 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.
- 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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