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: \[ -\triangle u +\operatorname {div}(u\mathbf {b}) =f \quad \text { and }\quad -\triangle v -\mathbf {b} \cdot \nabla v =g \] 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 [J. Funct. Anal. 130 (1995), pp. 161–219] and Fabes-Mendez-Mitrea [J. Funct. Anal. 159 (1998), pp. 323–368] 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 $n=2$.