Scalar elliptic equations with a singular drift

Abstract

We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-\Delta u+b^{(\alpha )}\cdot \nabla u=f$ in a bounded domain $\Omega \subset {\mathbb{R}}^2$ containing the origin, where $f\in W_q^{-1}(\Omega )$ with $q>2$ and $b^{(\alpha )}:=b-\alpha \frac{x}{|x{|}^2}$, $b$ is a divergence-free vector field and $\alpha \in \mathbb{R}$ is a parameter.

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