Harnack inequalities on a manifold with positive or negative Ricci curvature
type: Item
instance of: scholarly article
title: Harnack inequalities on a manifold with positive or negative Ricci curvature
zbMATH Open document ID: 0924.58096
DOI: 10.4171/RMI/253
author: Dominique Bakry
author: Zhongmin Qian
published in: Revista Matemática Iberoamericana
publication date: 1999-11-08T00:00:00Z
full work available at URL: https://eudml.org/doc/39567
review text: The aim of the authors is to present new Harnack estimates for non-negative solutions of the heat equation on a complete Riemannian manifold of dimension \(n\) with Ricci curvature bounded by a constant. Let us mention that this constant may be positive or negative. Notice that the method used by the authors is the maximum principle, which plays a fundamental role in the theory of partial differential equations, and the Bochner identity in some nice test functions (this has been studied in several papers, for instance by S. T. Yan, S. Y. Chang and P. Li). The crucial point is that any family of test functions gets different kinds of results; the test functions in use are related to the results one is looking for. In general, it is quite hard to decide what is the best estimate one could expect from a given inequality. The authors in this note develop a method which produces the best possible estimates and show how to construct good test functions useful to prove the estimates requested using the maximum principle.
reviewed by: Maria Alessandra Ragusa
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 1275490
zbMATH Keywords: Harnack inequality
zbMATH Keywords: Ricci curvature
zbMATH Keywords: heat equations
zbMATH Keywords: Riemann manifold
zbMATH Keywords: maximum principle
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1275490