The Dirichlet Problem for elliptic equations with singular drift terms

Abstract

We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $\Omega$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla u) + {\bf B}\cdot \nabla u=:L_0 u+ {\bf B}\cdot \nabla u=0, \] given that the analogous result holds (typically with a different value of $p$) for the homogeneous second order operator $L_0$. Essentially, we assume that $|\mathbf{B}(X)|\lesssim \text{dist}(X,\partial \Omega {)}^{-1}$, and that $|\mathbf{B}(X){|}^2\text{dist}(X,\partial \Omega )dX$ is a Carleson measure in $\Omega$.

New version corrects an error in the proof of estimate (7.6). The author is grateful to Murat Akman for pointing out the error