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 $\mathrm{\Omega }$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form $$Lu=-\text{div}(A\mathrm{\nabla }u)+\mathbf{B}\cdot \mathrm{\nabla }u=:L_{0}u+\mathbf{B}\cdot \mathrm{\nabla }u=0,$$ 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 $\mathrm{\mid }\mathbf{B}(X)\mathrm{\mid }\lesssim \text{dist}(X,\mathrm{\partial }\mathrm{\Omega }{)}^{-1}$, and that $\mathrm{\mid }\mathbf{B}(X){\mathrm{\mid }}^2\text{dist}(X,\mathrm{\partial }\mathrm{\Omega })dX$ is a Carleson measure in $\mathrm{\Omega }$.