Solvability of the Neumann Problem for Elliptic Equations in Chord-Arc Domains with Very Big Pieces of Good Superdomains

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