Oscillations and concentrations generated by ${\mathcal A}$-free mappings and weak lower semicontinuity of integral functionals
Irene Fonseca, Martin Kružík · ESAIM Control Optimisation and Calculus of Variations · 2009
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”.