OSCILLATIONS AND CONCENTRATIONS GENERATED BY A-FREE MAPPINGS AND WEAK LOWER SEMICONTINUITY OF INTEGRAL FUNCTIONALS

摘要

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps {u_k}_(k∈N) is contained in L~p(Ω;R~m) satisfying a linear differential constraint Au_k = 0. Applications to sequential weak lower semicontinuity of integral functional 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 ▽_(ψ_k)- det ▽_ψ in measures on the closure of Ω is contained in R~n if ψ_k→ ψ in W~(1,n) (Ω;R~n). This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma precisely stating which subsets Ω_j is contained in Ω must be removed to obtain weak lower semicontinuity of u → ∫_(ΩΩ_j) v(u(x))dx along {u_k} is contained in L~p(Ω;R~m)∩ ker A. Specifically, Ω_j are arbitrarily thin "boundary layers".