Oscillations and concentrations generated by -free mappings and weak lower semicontinuity of integral functionals

¹ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. fonseca@andrew.cmu.edu
² Institute of Information Theory and Automation of the ASCR, 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

Received: 3 October 2008
Revised: 9 December 2008

Abstract

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps satisfying a linear differential constraint . Applications to sequential weak lower semicontinuity of integral functionals on -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det in measures on the closure of if in . This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma precisely stating which subsets must be removed to obtain weak lower semicontinuity of along . Specifically, are arbitrarily thin “boundary layers”.