Gradient and Harnack inequalities on noncompact manifolds with boundary

Feng-Yu Wang

Vol. 245 (2010), No. 1, 185–200

DOI: 10.2140/pjm.2010.245.185

Abstract

By using the reflecting diffusion process and a conformal change of metric, a generalized maximum principle is established for (unbounded) time-space functions on a class of noncompact Riemannian manifolds with (nonconvex) boundary. As applications, Li–Yau-type gradient and Harnack inequalities are derived for the Neumann semigroup on a class of noncompact manifolds with (nonconvex) boundary. These generalize some previous ones obtained for the Neumann semigroup on compact manifolds with boundary. As a byproduct, the gradient inequality for the Neumann semigroup derived by Hsu on a compact manifold with boundary is confirmed on these noncompact manifolds.

Keywords

gradient estimate, Harnack inequality, generalized maxium principle

Mathematical Subject Classification 2000

Primary: 58J35, 60J60

Milestones

Received: 13 November 2008

Revised: 25 August 2009

Accepted: 3 December 2009

Published: 20 January 2010