Elliptic equations in divergence form with drifts in $L^2$

Abstract

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+\lambda u=f+\mathrm{div } \mathbf{F}\quad \text{in } \Omega\quad\text{and}\quad u=0\quad \text{on } \partial\Omega, \end{equation*} in bounded Lipschitz domain $\Omega$ in $\mathbb{R}^2$, where $A:\mathbb{R}^2\rightarrow \mathbb{R}^{2^2}$, $\mathbf{b} : \Omega\rightarrow \mathbb{R}^2$, and $\lambda \geq 0$ are given. If $2<p<\infty$ and $A$ has a small mean oscillation in small balls, $\Omega$ has small Lipschitz constant, and $\mathrm{div } A,\,\mathbf{b} \in L^{2}(\Omega;\mathbb{R}^2)$, then we prove existence and uniqueness of weak solutions in $W^{1,p}_0(\Omega)$ of the problem. Similar result also holds for the dual problem.