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

Let Ω ⊂ Rⁿ⁺¹ be a bounded chord-arc domain, let L=- div A∇ be an elliptic operator in Ω, and let 1<p≤ 2. In this paper we show that if the regularity problem for L is solvable in Lq for some $q>p$ in Ω, ∂ Ω supports a weak p-Poincar\'e inequality, and Ω has very big pieces of superdomains for which the Neumann problem for L is solvable uniformly in Lq, then the Neumann problem for L is solvable in Lᵖ in Ω.