Inserted: 3 mar 2009
Last Updated: 10 feb 2015
Journal: Publ. Mat.
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