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 \(Ω\) (i.e., a uniform domain with Ahlfors–David regular boundary), for elliptic equations of the form \[Lu=-div(A∇ u) +B· ∇ u=:L_0 u+ P· ∇ u=0,\] 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 \(|B(X)| dist(X,∂Ω)⁻¹\), and that \[|B(X)|^2dist(X,∂Ω)\, dX\] is a Carleson measure in \(Ω\).