Title: SHARP GRADIENT ESTIMATE AND YAU'S LIOUVILLE THEOREM FOR THE HEAT EQUATION ON NONCOMPACT MANIFOLDS
Abstract: Bulletin of the London Mathematical SocietyVolume 38, Issue 6 p. 1045-1053 Articles Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds Philippe Souplet, Philippe Souplet Laboratoire Analyse, Géométrie et Applications, Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, France e-mail: [email protected] for more papers by this authorQi S. Zhang, Qi S. Zhang Department of Mathematics, University of California, Riverside, CA 92521, USA e-mail: [email protected] for more papers by this author Philippe Souplet, Philippe Souplet Laboratoire Analyse, Géométrie et Applications, Institut Galilée, Université Paris-Nord, 93430 Villetaneuse, France e-mail: [email protected] for more papers by this authorQi S. Zhang, Qi S. Zhang Department of Mathematics, University of California, Riverside, CA 92521, USA e-mail: [email protected] for more papers by this author First published: 23 December 2016 https://doi.org/10.1112/S0024609306018947Citations: 104AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat 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 Rn without such a condition. We also prove a sharpened long-time gradient estimate for the log of the heat kernel on noncompact manifolds. 2000 Mathematics Subject Classification 35K05, 58J35. Citing Literature Volume38, Issue6December 2006Pages 1045-1053 RelatedInformation