Oscillations and concentrations in sequences of gradients

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 věží 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: 18 July 2006

Abstract

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, , bounded in if p > 1 and is a bounded domain with the extension property in . 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 Ω are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.