Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains

Abstract

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^n$ $(n\geq 3)$, where $\mathbf{b}:\Omega \rightarrow \mathbb{R}^n$ is a given vector field. Under the assumption that $\mathbf{b} \in L^{n}(\Omega)^n$, we first establish existence and uniqueness of solutions in $L_{\alpha}^{p}(\Omega)$ for the Dirichlet and Neumann problems. Here $L_{\alpha}^{p}(\Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(\alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig [17] and Fabes-Mendez-Mitrea [12] for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(\partial\Omega)$. Our results for the Dirichlet problems hold even for the case $n=2$.