Some cases of homogenization with unbounded oscillating constraints on the gradient

Abstract

For every bounded open set Ω with Lipschitz boundary, β in L^∞(Ω) we study the asymptotic behaviour (as h→∞) of the sequence i_h(Ω,β)=inf {∫_Ωf(hx,Du)+∫_Ωβu, u Lipschitz continuous, u=0 on ∂Ω,|Du(x)|≤ϕ(hx) for a.e. x in Ω} where f is a nonnegative function on Rⁿ×Rⁿ measurable and ]0,1[ⁿ-periodic in the x variable, convex in the z variable, ϕ is a ]0,1[ⁿ-periodic function from Rⁿ into [0,+∞] such that there exist Θ∈[0,½[, m>0 with 0<m≤ϕ(x) for a.e. x in ]0,1[ⁿ−]½−Θ, ½+Θ[ⁿ, and one of the following conditions hold: f|z|^p≤f(x,z), p>n; or ϕ∈L^p(]0,1[ⁿ) p>n. It is proved that i_h(Ω,β) converges to a minimim problem of integral type for which an explicit formula is proved.