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SHARP GRADIENT ESTIMATE AND YAU'S LIOUVILLE THEOREM FOR THE HEAT EQUATION ON NONCOMPACT MANIFOLDS

Published online by Cambridge University Press:  19 December 2006

PHILIPPE SOUPLET  and
QI S. ZHANG
PHILIPPE SOUPLET
Affiliation:
Laboratoire Analyse, Géométrie et Applications, Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, Francesouplet@math.univ-paris13.fr
QI S. ZHANG
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, USAqizhang@math.ucr.edu
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Abstract

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related to the Cheng–Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau's celebrated Liouville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Liouville theorem for the heat equation does not hold even in ${\mathbb R}^n$ without such a condition. We also prove a sharpened long-time gradient estimate for the log of the heat kernel on noncompact manifolds.

Keywords

35K0558J35
Type
Papers
Information
Bulletin of the London Mathematical Society , Volume 38 , Issue 6 , December 2006 , pp. 1045 - 1053
Copyright
The London Mathematical Society 2006

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