Oscillations and concentrations generated by Afree mappings and weak low semicontinuity of integral functions

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;Rm) 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∇ϕk ∗ ⇀ det∇ϕ in measures on the closure of Ω ⊂ Rn if ϕk ⇀ ϕ in W 1,n(Ω;Rn). 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} ⊂ L p(Ω;Rm)∩kerA. Specifically, Ωj are arbitrarily thin “boundary layers”