Existence, uniqueness, and regularity results for elliptic equations with drift terms in critical weak spaces

Abstract

We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $\Omega$ in ${\mathbb{R}}^n$, $n\ge 3$, with drifts $\mathbf{b}$ in the critical weak $L^n$-space $L^{n,\infty }(\Omega ;{\mathbb{R}}^n)$. First, assuming that the drift $\mathbf{b}$ has nonnegative weak divergence in $L^{n/2,\infty }(\Omega )$, we establish existence and uniqueness of weak solutions in $W^{1,p}(\Omega )$ or $D^{1,p}(\Omega )$ for any $p$ with $n^{\prime }=n/(n-1)<p<n$. By duality, a similar result also holds for the dual problem. Next, we prove $W^{1,n+\epsilon }$ or $W^{2,n/2+\delta }$-regularity of weak solutions of the dual problem for some $\epsilon ,\delta >0$ when the domain $\Omega$ is bounded. By duality, these results enable us to obtain a quite general uniqueness result as well as an existence result for weak solutions belonging to ${\bigcap }_{p<n^{\prime }}{W}^{1,p}(\Omega )$. Finally, we prove a uniqueness result for exterior problems, which implies in particular that (very weak) solutions are unique in both $L^{n/(n-2),\infty }(\Omega )$ and $L^{n,\infty }(\Omega )$.