### 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 language | English (US) |
---|---|

Pages (from-to) | 1-36 |

Number of pages | 36 |

Journal | Annales Scientifiques de l'Ecole Normale Superieure |

Volume | 42 |

Issue number | 1 |

State | Published - Jan 2009 |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

*Annales Scientifiques de l'Ecole Normale Superieure*, vol. 42, no. 1, pp. 1-36.

}

TY - JOUR

T1 - Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula

AU - Kotschwar, Brett

AU - Ni, Lei

PY - 2009/1

Y1 - 2009/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77953913302&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953913302&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77953913302

VL - 42

SP - 1

EP - 36

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

SN - 0012-9593

IS - 1

ER -