Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula
By Brett Kotschwar, Lei Ni. 2007
138 citations recorded by the source corpus.
In the first part of this paper, we prove local interior and boundary gradient estimates for p-harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the 1/H (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues of the p-Laplace equation and prove sharp Li-Yau type gradient estimates for positive solutions to these equations on manifolds of nonnegative Ricci curvature. For one of these equations, we also prove an entropy monotonicity formula generalizing an earlier such formula of the second author for the linear heat equation. As an application of this formula, we show that a complete Riemannian manifold with non-negative Ricci curvature and sharp L^p-logarithmic Sobolev inequality must be isometric to Euclidean space.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/7d73fb641664846aa385a7f9545cf9576056d651
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
- Mattia Fogagnolo
- L. Mazzieri
- Virginia Agostiniani
- Xiaodong Wang
- Xiang-Dong Li
- Lei Zhang
- Zoltán M. Balogh
- Alexandru Krist'aly
In Geometry and Topology
- L. Mazzieri
- Virginia Agostiniani
- Mattia Fogagnolo
- Xiaodong Wang
- Xiang-Dong Li
- A. Neves
- Kevin Brighton
- Guangyue Huang
Citations in the corpus
- Minkowski Inequalities via Nonlinear Potential Theory (2019, background): “Moser [51] and subsequently extended by Kotschwar and Ni [39] (see also the very recent [48]).”
- Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature (2012, background): “Since Perelman’s preprint [33] was published on Arxiv in 2002, many people have extended the monotonicity of the W-entropy to other type geometric heat flows on Riemannian manifolds [13, 20 ,25,29,30].”
- Local gradient estimate for p-harmonic functions on Riemannian manifolds (2010, background): “In particular R. Moser [11] established a nice connection between p-harmonic functions and the inverse mean curvature flow.”