ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

ESAIM: COCV, Vol. 14, N°1, pp. 71-104
DOI: 10.1051/cocv:2007051

Oscillations and concentrations in sequences of gradients

Agnieszka Kalamajska¹ and Martin Kruzík^{2, 3}

¹ Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland; Agnieszka.Kalamajska@mimuw.edu.pl
² Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic. Corresponding address Pod vodárenskou vezí 4, 182 08 Praha 8, Czech Republic.
³ Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29 Praha 6, Czech Republic; kruzik@utia.cas.cz (corresponding author).

(Received July 18, 2006. Published online September 21, 2007.)

Abstract
We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\{\nabla u_k\}$ , bounded in $L^p(\Omega;{\mathbb R}^{m\times n})$ if p > 1 and $\Omega\subset{\mathbb R}^n$ is a bounded domain with the extension property in W^1,p. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega$ are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.

Mathematics Subject Classification. 49J45, 35B05

Key words: Sequences of gradients, concentrations, oscillations, quasiconvexity

© EDP Sciences, SMAI 2007