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

Kotschwar, Brett; Ni, Lei

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 1, pp. 1-36.

Dans la première partie de cet article, nous établissons des estimées locales de gradient pour les fonctions $p$-harmoniques à l’intérieur et au bord, sur les variétés riemanniennes générales. Grâce à ces estimations et suivant une idée récente de R. Moser, nous obtenons un théorème d’existence de solutions faibles au sens de la formulation d’ensemble de niveau pour le flot $1/H$ (inverse de la courbure moyenne) des hypersurfaces dans les variétés ambiantes ayant la propriété de la croissance optimale du volume. Dans la deuxième partie, nous considérons deux types d’équations paraboliques pour les fonctions $p$-harmoniques et nous établissons une estimation optimale du type de Li-Yau pour les solutions positives pour ces équations sur les variétés à courbure de Ricci non-négative. Nous montrons aussi une formule de monotonie des entropies associées à ces équations. Cette formule généralise un résultat antérieur du deuxième auteur pour l’équation de la chaleur linéaire. Comme application, nous montrons que toute variété riemannienne complète à courbure de Ricci positive ou nulle et admettant une inégalité logarithmique $L^p$ optimale est isométrique à l’espace euclidien.

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.

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