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

TL;DR

This paper establishes conditions under which the Neumann problem for elliptic equations in chord-arc domains is solvable in $L^p$, given solvability in larger subdomains and certain geometric and regularity conditions.

Contribution

It provides a new solvability transfer result for the Neumann problem in complex domains with big superdomains, under minimal regularity assumptions.

Findings

Neumann problem solvable in $L^p$ under specified conditions

Solvability in larger superdomains implies solvability in the original domain

Requires the domain to support a weak $p$-Poincaré inequality

Abstract

Let $\Omega \subset \mathbb{R}^{n+1}$ be a bounded chord-arc domain, let $\mathcal L=-{\rm div} A\nabla$ be an elliptic operator in $\Omega$ associated with a matrix $A$ having Dini mean oscillation coefficients, and let $1<p\leq 2$. 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 $p$-Poincar\'e 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 $\Omega$.