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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$ be a bounded chord-arc domain, let $\mathcal{L}=-\mathrm{d}\mathrm{i}\mathrm{v}A\mathrm{\nabla }$ be an elliptic operator in $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$, $\mathrm{\partial }\mathrm{\Omega }$ supports a weak $p$-Poincaré inequality, and $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$.