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# Communications in Mathematical Sciences

## Volume 12 (2014)

### Number 4

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

Pages: 681 – 694

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n4.a4

#### Authors

#### Abstract

We consider an elliptic equation with a divergence-free drift b. We prove that an inequality of Harnack type holds under the assumption $b \in L^{n / 2+\delta}$ where $\delta \gt 0$. As an application we provide a one-sided Liouville’s theorem provided that $b \in L^{n / 2+\delta}(\mathbb{R}^n)$.

#### Keywords

Harnack inequality, Liouville theorem, regularity, drift-diffusion equations

#### 2010 Mathematics Subject Classification

35B53, 35B65, 35J15, 35Q35