Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$ be a bounded chord-arc domain, let $\mathcal{L}=-\mathrm{div}A\nabla$ be an elliptic operator in $\Omega$ associated with a matrix $A$ having Dini mean oscillation coefficients, and let $1<p\le 2$. In this paper we show that if the regularity problem for $\mathcal{L}$ is solvable in $L^q$ for some $q>p$ in $\Omega$, $\partial \Omega$ supports a weak $p$-Poincaré inequality, and $\Omega$ has very big pieces of superdomains for which the Neumann problem for $\mathcal{L}$ is solvable uniformly in $L^q$, then the Neumann problem for $\mathcal{L}$ is solvable in $L^p$ in $\Omega$.

Minor corrections. To appear in Archive for Rational Mechanics and Analysis (ARMA)