Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula

Research output: Contribution to journalArticle

43 Citations (Scopus)

Abstract

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-harmonic 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 nonnegative Ricci curvature and sharp L p-logarithmic Sobolev inequality must be isometric to Euclidean space.

Original languageEnglish (US)
Pages (from-to)1-36
Number of pages36
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Volume42
Issue number1
StatePublished - Jan 2009
Externally publishedYes

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P-harmonic
Gradient Estimate
Harmonic Functions
Nonnegative Curvature
Entropy
Ricci Curvature
Riemannian Manifold
Monotonicity Formula
Volume Growth
Logarithmic Sobolev Inequality
Mean Curvature Flow
Isometric
Level Set
Heat Equation
Existence Theorem
Weak Solution
Hypersurface
Euclidean space
Positive Solution
Linear equation

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula. / Kotschwar, Brett; Ni, Lei.

In: Annales Scientifiques de l'Ecole Normale Superieure, Vol. 42, No. 1, 01.2009, p. 1-36.

Research output: Contribution to journalArticle

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