Harnack inequality for nondivergent elliptic operators on Riemannian manifolds

Seick Kim

Vol. 213 (2004), No. 2, 281–293

DOI: 10.2140/pjm.2004.213.281

Abstract

We consider second-order linear elliptic operators of nondivergence type which are intrinsically defined on Riemannian manifolds. Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabré’s result and, as a consequence, we give another proof to the Harnack inequality of Yau for positive harmonic functions on Riemannian manifolds with nonnegative Ricci curvature using the nondivergence structure of the Laplace operator.

Milestones

Received: 2 August 2002

Revised: 23 January 2003

Published: 1 February 2004