In this paper we settle (in dimension $n=2$) the open questionwhether for a divergence form equation $\div (A\nabla u) = 0$ withcoefficients satisfying certain minimal smoothness assumption (a Carleson measure condition), the $L^p$ Neumann and Dirichlet regularityproblems are solvable for some values of $p\in (1,\infty)$. The related question for the $L^p$ Dirichlet problem was settled (in any dimension) in 2001 by Kenig and Pipher [Kenig, C.E. and Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45 (2001), no. 1, 199-217].
"Elliptic equations in the plane satisfying a Carleson measure condition." Rev. Mat. Iberoamericana 26 (3) 1013 - 1034, September, 2010.