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Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds
Author(s) -
Souplet Philippe,
Zhang Qi S.
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609306018947
Subject(s) - mathematics , heat equation , heat kernel , bounded function , mathematical analysis , laplace's equation , laplace transform , harmonic function , elliptic curve , pure mathematics , partial differential equation
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 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. 2000 Mathematics Subject Classification 35K05, 58J35.