The Dirichlet problem for elliptic equations with singular drift terms

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
\[Lu=-\operatorname{div}(A\nabla u) +\mathbf{B}\cdot \nabla u=:L_0 u+ \mathbf{P}\cdot \nabla u=0,\]
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 \(|\mathbf{B}(X)|\lesssim \operatorname{dist}(X,\partial\Omega)^{-1}\), and that
\[|\mathbf{B}(X)|^2\operatorname{dist}(X,\partial\Omega)\, dX\]

is a Carleson measure in \(\Omega\).