Harnack inequalities on a manifold with positive or negative Ricci curvature
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.
scientific article; zbMATH DE number 1275490