Multiple integrals under differential constraints: two-scale convergence and homogenization

Abstract

Two-scale techniques are developed for sequences of maps {u k } ⊂ L p (Ω; ℝ M ) satisfying a linear differential constraint Au k = 0. These, together with Γ-convergence arguments and using the unfolding operator, provide a homogenization result for energies of the type F e (u) := ∫ Ω f(x,x/e, u(x)) d x with u ∈ L p (Ω; ℝ M ), Au = 0, that generalizes current results in the case where A = curl.