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

Hyunseok Kim, Hyunwoo Kwon

Published 2018-11-30Version 1

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 (1995, JFA) and Fabes-Mendez-Mitrea (1998, JFA) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(\partial\Omega)$.