Nonlinear reinforcement problems with right-hand side in L 1

Abstract

We study the asymptotic behaviour, as ε goes to zero, of the entropy solution u^ε to a class of nonlinear elliptic problems whose prototype is

−div((δ^ε)^p−1|∇u^ε|^p−2∇u^ε)=f^ε

in Ω^ε, u^ε=0 on ∂Ω^ε, where 1<p<N, $\varOmega^{\varepsilon}=\overline{\varOmega}\cup\Sigma^{\varepsilon}$ , Ω⊂R^N is a bounded domain of class C^1,1, surrounded along its boundary by a layer Σ^ε of small thickness ε, δ^ε=1 in Ω, δ^ε=ε in Σ^ε, and f^ε belongs to L¹(Ω^ε). The solutions u^ε also satisfies natural transmission conditions on ∂Ω.