Existence and uniqueness of weak solution in $W^{1,2+\epsilon }$ for elliptic equation with drifts in weak-$L^n$ spaces

Abstract

We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$: \[ \text{(a) } -\operatorname{div}(A \nabla u)+\operatorname{div}(u\mathbf{b})=f,\quad \text{(b) } -\operatorname{div}(A^T \nabla v)-\mathbf{b} \cdot \nabla v =g \quad \text{in } Ω, \] where $\Omega$ is a bounded Lipschitz domain in ${\mathbb{R}}^n$, $n\ge 2$. Assuming that $\mathbf{b}\in L^{n,\infty }(\Omega {)}^n$ has non-negative weak divergence in $\Omega$, we establish existence and uniqueness of weak solution in $W_0^{1,2+\epsilon }(\Omega )$ of the problem (b) when $A$ is bounded and uniformly elliptic. As an application, we prove unique solvability of weak solution $u$ in ${\bigcap }_{q<2}{W}_0^{1,q}(\Omega )$ for the problem (a) for every $f\in {\bigcap }_{q<2}{W}^{-1,q}(\Omega )$.

arXiv comment15 pages ; References are added, some proofs are modified