Boundary value problems in Lipschitz domains for equations with drifts

In this work we establish solvability and uniqueness for the $D_2$ Dirichlet problem and the $R_2$ Regularity problem for second order elliptic operators $L=-\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }\cdot )+b\mathrm{\nabla }\cdot$ in bounded Lipschitz domains, where $b$ is bounded, as well as their adjoint operators $L^t=-{\rm div}(A^t\nabla\cdot)-{\rm div}(b,\cdot)$. The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on $\mathrm{d}\mathrm{i}\mathrm{v}b$ in order to obtain solvability. On the other hand, we show that $R_2$ for $L$ and $D_2$ for $L^t$ are uniquely solvable, assuming only that $A$ is Lipschitz continuous (and not necessarily symmetric) and $b$ is bounded.