# A stability result for nonlinear Neumann problems in Reifenberg flat domains in $\mathbb R^N$.

created by lemenant on 03 Mar 2009
modified on 10 Feb 2015

[BibTeX]

Published Paper

Inserted: 3 mar 2009
Last Updated: 10 feb 2015

Journal: Publ. Mat.
Year: 2011

Abstract:

In this paper we prove that if $\Omega_k$ is a sequence of Reifenberg-flat domains in $\mathbb R^N$ that converges to $\Omega$ for the complementary Hausdorff metric and if in addition the sequence $\Omega_k$ has a uniform size of holes'', then the solutions $u_k$ of a Neumann problem of the divergence form converge to the solution $u$ of the same Neumann problem in $\Omega$. The result is obtained by proving the Mosco convergence of some Banach spaces. As an application, in the second part of the paper we prove a decay estimate on the gradient for solutions of nonlinear Neumann problems. The estimate is initially established when the boundary is flat and then a similar estimate for perturbed boundaries using the stability property is obtained.