The Boundary value problems for second order elliptic operators satisfying a Carleson condition

Abstract

Let $\Omega$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in $H^{1,p}(\partial\Omega)$ and of the Neumann problem with $L^p(\partial\Omega)$ data for the operator $L$ on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator $L$ to be rough obeying a certain Carleson condition with small norm. These results complete the results of [5] where $L^p(\partial\Omega)$ Dirichlet problem was considered under the same assumptions and [6] where the regularity and Neumann problems were considered on two dimensional domains.