A stability result for nonlinear Neumann problems under boundary variations

In this paper we study, in dimension two, the stability of the solutions of some nonlinear elliptic equations with Neumann boundary conditions, under perturbations of the domains in the Hausdorff complementary topology. More precisely, for every bounded open subset Ω of ℝ2, we consider the problem: {-diva(x,∇uΩ)+b (x,uΩ)=0 in Ω, a (x,∇uΩ)·ν=0 on ∂Ω, where a:ℝ2×ℝ2→ℝ2 and b:ℝ2×ℝ→ℝ are two Carathéodory functions which satisfy the standard monotonicity and growth conditions of order p, with 1<p≤2. Let Ωn be a uniformly bounded sequence of open sets in ℝ2, whose complements Ωnc have a uniformly bounded number of connected components. We prove that, if Ωnc→Ωc in the Hausdorff metric and Ωn → Ω , then uΩn→u Ω and ∇uΩn →∇uΩ strongly in Lp. The proof is obtained by showing the Mosco convergence of the Sobolev spaces W1,p(Ωn) to the Sobolev space W1,p(Ω). © 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.