Some cases of homogenization with unbounded oscillating constraints on the gradient
Riccardo De Arcangelis, Antonio Vitolo · Asymptotic Analysis · 1992
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 {∫ Ω f(hx,Du)+∫ Ω βu, u Lipschitz continuous, u=0 on ∂Ω,|Du(x)|≤ϕ(hx) for a.e. x in Ω} where f is a nonnegative function on R n ×R n measurable and ]0,1[ n -periodic in the x variable, convex in the z variable, ϕ is a ]0,1[ n -periodic function from R n into [0,+∞] such that there exist Θ∈[0,½[, m>0 with 0 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.