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

Let $\mathrm{\Omega }$ be a Lipschitz domain in ${\mathbb{R}}^n$ $n\ge 2,$ and $L=div(A\mathrm{\nabla }\cdot )$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in $H^{1,p}(\mathrm{\partial }\mathrm{\Omega })$ and of the Neumann problem with $L^p(\mathrm{\partial }\mathrm{\Omega })$ 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 $L^p(\mathrm{\partial }\mathrm{\Omega })$ Dirichlet problem was considered under the same assumptions and [6] where the regularity and Neumann problems were considered on two dimensional domains.