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

We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $Ω$ in $\mathbb{R}^n$, $n \ge 3$, with drifts $\mathbf{b}$ in the critical weak $L^n$-space $L^{n,\infty}(Ω; \mathbb{R}^n )$. First, assuming that the drift $\mathbf{b}$ has nonnegative weak divergence in $L^{n/2, \infty }(Ω)$, we establish existence and uniqueness of weak solutions in $W^{1,p}(Ω)$ or $D^{1,p}(Ω)$ for any $p$ with $n' = n/(n-1)< p < n$. By duality, a similar result also holds for the dual problem. Next, we prove $W^{1,n+\varepsilon}$ or $W^{2, n/2+δ}$-regularity of weak solutions of the dual problem for some $\varepsilon, δ>0$ when the domain $Ω$ 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 $\bigcap_{p< n' }W^{1,p}(Ω)$. Finally, we prove a uniqueness result for exterior problems, which implies in particular that (very weak) solutions are unique in both $L^{n/(n-2),\infty}(Ω)$ and $L^{n,\infty}(Ω)$.

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