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+λu=f+\mathrm{div } \mathbf{F}\quad \text{in } Ω\quad\text{and}\quad u=0\quad \text{on } \partialΩ, \end{equation*} in bounded Lipschitz domain $\Omega$ in ${\mathbb{R}}^2$, where $A:{\mathbb{R}}^2\to {\mathbb{R}}^{2^2}$, $\mathbf{b}:\Omega \to {\mathbb{R}}^2$, and $\lambda \ge 0$ 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 $W_0^{1,p}(\Omega )$ of the problem. Similar result also holds for the dual problem.

arXiv comment15 pages; v2: title, abstracts are changed. Several comments were reflected / v3: typos are corrected / v4: final verison, accepted to Proc. AMS / v5: really minor typo on my name