  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 $\Omega$, 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 $\Omega$.
