The Dirichlet Problem for elliptic equations with singular drift terms

TL;DR

This paper proves the solvability of the Dirichlet problem for certain elliptic equations with singular drift terms in specific geometric domains, extending known results for related operators under Carleson measure conditions.

Contribution

It establishes $L^p$ solvability for elliptic equations with singular drift in 1-sided chord-arc domains, building on existing results for homogeneous operators.

Findings

$L^p$ solvability established for elliptic equations with singular drift

Conditions on drift term involve Carleson measure and decay near boundary

Extension of known results to more general elliptic operators

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 $|{\bf B}(X)|\lesssim \text{dist}(X,\partial \Omega)^{-1}$, and that $|{\bf B}(X)|^2\text{dist}(X,\partial \Omega) dX$ is a Carleson measure in $\Omega$.