Robin problems for elliptic equations with singular drifts on Lipschitz domains

Abstract

Let \(n\ge 2\) and \(\Omega \subset \mathbb {R}^n\) be a bounded Lipschitz domain. Assume that \(\textbf{b}\in L^{n*}(\Omega ;\mathbb {R}^n)\) and \(\gamma \) is a non-negative function on \(\partial \Omega \) satisfying some mild assumptions, where \(n^*:=n\) when \(n\ge 3\) and \(n^*\in (2,\infty )\) when \(n=2\). In this article, we establish the unique solvability of the Robin problems

$$\begin{aligned} \left\{ \begin{aligned} -\Delta u+\textrm{div}(u\textbf{b})&=f{} & {} \text {in}\ \ \Omega , \\ \left( \nabla u-u\textbf{b}\right) \cdot \varvec{\nu }+\gamma u&=u_R{} & {} \text {on}\ \ \partial \Omega \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{aligned} -\Delta v-\textbf{b}\cdot \nabla v&=g{} & {} \text {in}\ \ \Omega , \\ \nabla v\cdot \varvec{\nu }+\gamma v&=v_R{} & {} \text {on}\ \ \partial \Omega \end{aligned}\right. \end{aligned}$$

in the Bessel potential space \(L^p_\alpha (\Omega )\), where \(\alpha \in (0,2)\) and \(p\in (1,\infty )\) satisfy some restraint conditions, and \(\varvec{\nu }\) denotes the outward unit normal to the boundary \(\partial \Omega \). The results obtained in this article extend the corresponding results established by Kim and Kwon (Trans Am Math Soc 375:6537–6574, 2022) for the Dirichlet and the Neumann problems to the case of the Robin problem.