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 $\hat{I}\text{c◯}$ (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 $|\mathbf{B}(X)|\lesssim \text{dist}(X,\partial \hat{I}\text{c◯}{)}^{-1}$, and that $|\mathbf{B}(X){|}^2\text{dist}(X,\partial \hat{I}\text{c◯})dX$ is a Carleson measure in $\hat{I}\text{c◯}$.

New version corrects an historical error in the introduction, pertaining to the cited paper [HL], regarding the doubling property of elliptic-harmonic measure for operators with a drift. A new section (Section 7) has been added, with a proof of doubling following the argument in [HL]