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

Authors

Mihaela Ignatova (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Igor Kukavica (Department of Mathematics, University of Southern California, Los Angeles, Calif., U.S.A.)

Lenya Ryzhik (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

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

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Published 7 February 2014