The Harnack inequality for second-order elliptic equations with divergence-free drifts

Abstract

We consider an elliptic equation with a divergence-free drift b. We prove that an inequality of Harnack type holds under the assumption b ∈ L ∩L where δ > 0. As an application we provide a one sided Liouville’s theorem provided that b ∈ L(R) ∩ L(R).