Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds
By Philippe Souplet, Qi S. Zhang. 2005
236 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/9482f392db5d60e865d47e0be56fa9ddbbc31765
How researchers see this paper
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In Applied Mathematics
- Xiaodong Cao
- Qi S. Zhang
- Juan Luis V'azquez
- Lei Ni
- Mihai Băileşteanu
- Artem Pulemotov
- Cédric Villani
- José Antonio Carrillo
In Geometry and Topology
- Qi S. Zhang
- Xiaodong Cao
- Lei Ni
- Mihai Băileşteanu
- Artem Pulemotov
- Juan Luis V'azquez
- P. Lu
- Cédric Villani
Citations in the corpus
- Some gradient estimates for the heat equation on domains and for an equation by Perelman (2006, methodology): “Remark. As pointed out in [ SZ ], the local and global estimate can not replace each other.”
- The Li-Yau inequality and applications under a curvature-dimension condition (2014, unknown): “Let us mention in particular the most recent [15, 19, 4, 12, 9, 14, 7, 13].”
- Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds (2008, result): “Our local estimate is more closely related to the result of Li–Yau mentioned in the introduction, than that of say [29].”
- Gradient estimates for the heat equation under the Ricci flow (2009, background): “Hamilton’s result were proposed in [31, 36, 5, 6].”
- L2 curvature bounds on manifolds with bounded Ricci curvature (2016, unknown): “53001 [27] Souplet, Philippe; Zhang, Qi S.”