Boundary Value Problems for Elliptic Operators Satisfying Carleson Condition

In this note, we present in concise form recent results on solvability of the $$L^p$$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying certain natural of Carleson condition. More precisely, with $$L=\mbox{div}(A\nabla )$$ , we assume the matrix A is elliptic $$\mbox{dist}(X,\partial \Omega )^{-1}\left (\mbox{osc}_{B(X,\delta (X)/2)}A\right )^2\,dX$$ is a Carleson measure. We present two types of results, the so-called “small Carleson” case where, for a given $$1