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

Brett Kotschwar; Lei Ni

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 1, page 1-36
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Kotschwar, Brett, and Ni, Lei. "Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula." Annales scientifiques de l'École Normale Supérieure 42.1 (2009): 1-36. <http://eudml.org/doc/272232>.

@article{Kotschwar2009,
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.},
author = {Kotschwar, Brett, Ni, Lei},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {$p$-harmonic functions; inverse mean curvature flow; entropy monotonicity formula},
language = {eng},
number = {1},
pages = {1-36},
publisher = {Société mathématique de France},
title = {Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula},
url = {http://eudml.org/doc/272232},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Kotschwar, Brett
AU - Ni, Lei
TI - Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 1
SP - 1
EP - 36
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.
LA - eng
KW - $p$-harmonic functions; inverse mean curvature flow; entropy monotonicity formula
UR - http://eudml.org/doc/272232
ER -

References

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