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*} -div(A\nabla u)+\mathbf {b} \cdot \nabla u+\lambda u=f+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 $divA,\,\mathbf {b} \in L^{2}(\Omega ;\mathbb {R}^2)$, 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.