*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.

**Keywords:**
stability, Mosco convergence, Reifenferg flat, Neumann Sieve

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