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 $L=-{\rm div}(A\nabla\cdot)+b\nabla\cdot$ in bounded Lipschitz domains, for which $b$ 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 methods applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on $\dive b$ 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, only assuming that $A$ is Lipschitz continuous (and not necessarily symmetric) and $b$ is just bounded.