A Refined Subsolution Estimate of Weak Subsolutions to Second Order Linear Elliptic Equations with a Singular Vector Field

Abstract

We consider second order linear elliptic equations $ - \div (A(x) \nabla u) + \mathbf{b}(x) \cdot \nabla u = 0$ with a singular vector field $\mathbf{b}$. We prove a refined subsolution estimate, which contains a precise dependence of the quantities of $\mathbf{b}$, for weak subsolutions and a weak Harnack inequality for weak supersolutions under certain assumptions on $\mathbf{b}$.