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

Abstract

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps {uk} k∈N ⊂ Lp(Ω;RDm) satisfying a linear differential constraint Auk=0. Applications to sequential weak lower semicontinuity of integral functionals on A-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det ∇φ k⇀*det∇φ in measures on the closure of Ω⊂RDn if φk⇀φ in W 1,n(ΩRDn). This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma precisely stating which subsets Ωj⊂Ω; must be removed to obtain weak lower semicontinuity of u→∫ Ω\Ωj} v(u(x))dx along {uk}⊂ Lp(Ω;RDm)∩ker A. Specifically, Ωj are arbitrarily thin "boundary layers". © EDP Sciences, SMAI, 2009.