Lower‐semicontinuity result for the two‐scale convergence
Mahdi Boukrouche, Ionel Sorin Ciuperca · PAMM · 2007
Abstract Let (m, n) ∈ ℕ2, Ω an open bounded domain in ℝm , Y = [0, 1]m ; uε in (L2(Ω))n which is two‐scale converges to some u in (L2(Ω × Y))n . Let φ: Ω × ℝm × ℝn → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝm × ℝn , φ(x, ·, z) is 1‐periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C1 > 0 and a function C2 ∈ L2(Ω) such that magnified image Then we have magnified image (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)