Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula  [ Estimées de gradient locales, de fonctions p-harmoniques, 1/H flot et une formule d’entropie ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 1, p. 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.
DOI : https://doi.org/10.24033/asens.2089
Classification:  35J60,  53C44
Mots clés: fonctions p-harmoniques, flot de l’inverse de la courbure moyenne, formule monotone de l’entropie
@article{ASENS_2009_4_42_1_1_0,
     author = {Kotschwar, Brett and Ni, Lei},
     title = {Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {1},
     year = {2009},
     pages = {1-36},
     doi = {10.24033/asens.2089},
     zbl = {1182.53060},
     mrnumber = {2518892},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2009_4_42_1_1_0}
}
Kotschwar, Brett; Ni, Lei. Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 1, pp. 1-36. doi : 10.24033/asens.2089. https://www.numdam.org/item/ASENS_2009_4_42_1_1_0/

[1] D. Bakry, D. Concordet & M. Ledoux, Optimal heat kernel bounds under logarithmic Sobolev inequalities, ESAIM Probab. Statist. 1 (1995/97), 391-407. | Numdam | MR 1486642 | Zbl 0898.58052

[2] G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 679-698. | MR 52948 | Zbl 0047.19204

[3] W. Beckner, Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math. 11 (1999), 105-137. | MR 1673903 | Zbl 0917.58049

[4] S. Bobkov, A functional form of the isoperimetric inequality for the Gaussian measure, J. Funct. Anal. 135 (1996), 39-49. | MR 1367623 | Zbl 0838.60013

[5] S. Y. Cheng & S.-T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333-354. | MR 385749 | Zbl 0312.53031

[6] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo & L. Ni, The Ricci flow: techniques and applications. Part I: Geometric aspects, Mathematical Surveys and Monographs 135, Amer. Math. Soc., 2007. | MR 2302600 | Zbl 1157.53034

[7] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo & L. Ni, The Ricci flow: techniques and applications. Part II: Analytic aspects, Mathematical Surveys and Monographs 144, Amer. Math. Soc., 2008. | MR 2365237 | Zbl 1157.53035

[8] M. Del Pino & J. Dolbeault, The optimal Euclidean L p -Sobolev logarithmic inequality, J. Funct. Anal. 197 (2003), 151-161. | MR 1957678 | Zbl 1091.35029

[9] M. Del Pino, J. Dolbeault & I. Gentil, Nonlinear diffusions, hypercontractivity and the optimal L p -Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl. 293 (2004), 375-388. | MR 2053885 | Zbl 1058.35124

[10] K. Ecker, Logarithmic Sobolev inequalities on submanifolds of Euclidean space, J. reine angew. Math. 522 (2000), 105-118. | MR 1758578 | Zbl 0952.46021

[11] J. R. Esteban & J. L. Vázquez, Homogeneous diffusion in 𝐑 with power-like nonlinear diffusivity, Arch. Rational Mech. Anal. 103 (1988), 39-80. | MR 946969 | Zbl 0683.76073

[12] J. R. Esteban & J. L. Vázquez, Régularité des solutions positives de l’équation parabolique p-laplacienne, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 105-110. | MR 1044625 | Zbl 0708.35043

[13] L. C. Evans, Entropy and partial differential equations, lecture notes at UC Berkeley.

[14] L. C. Evans & H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 1-20. | Numdam | MR 781589 | Zbl 0601.60076

[15] M. Feldman, T. Ilmanen & L. Ni, Entropy and reduced distance for Ricci expanders, J. Geom. Anal. 15 (2005), 49-62. | MR 2132265 | Zbl 1071.53040

[16] I. Gentil, The general optimal L p -Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations, J. Funct. Anal. 202 (2003), 591-599. | MR 1990539 | Zbl 1173.35424

[17] R. E. Greene & H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math. 699, Springer, 1979. | MR 521983 | Zbl 0414.53043

[18] R. S. Hamilton, The formation of singularities in the Ricci flow, in Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995, 7-136. | MR 1375255 | Zbl 0867.53030

[19] B. Hein, A homotopy approach to solving the inverse mean curvature flow, Calc. Var. Partial Differential Equations 28 (2007), 249-273. | MR 2284568 | Zbl 1105.35055

[20] I. Holopainen, Volume growth, Green's functions, and parabolicity of ends, Duke Math. J. 97 (1999), 319-346. | MR 1682233 | Zbl 0955.31003

[21] G. Huisken & T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), 353-437. | MR 1916951 | Zbl 1055.53052

[22] J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), 849-858. | MR 721568 | Zbl 0554.35048

[23] P. Li & L.-F. Tam, Green's function, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), 277-318. | MR 1331970 | Zbl 0827.53033

[24] P. Li & J. Wang, Complete manifolds with positive spectrum. II, J. Differential Geom. 62 (2002), 143-162. | MR 1987380 | Zbl 1073.58023

[25] P. Li & S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201. | MR 834612 | Zbl 0611.58045

[26] R. Moser, The inverse mean curvature flow and p-harmonic functions, J. Eur. Math. Soc. 9 (2007), 77-83. | MR 2283104 | Zbl 1116.53040

[27] L. Ni, The entropy formula for linear heat equation, J. Geom. Anal. 14 (2004), 87-100; addenda J. Geom. Anal. 14 (2004), 369-374. | MR 2030576 | Zbl 1062.58028

[28] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint arXiv:math.DG/0211159. | Zbl 1130.53001

[29] J. L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type, Oxford Lecture Series in Mathematics and its Applications 33, Oxford University Press, 2006. | MR 2282669 | Zbl 1113.35004

[30] S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. | MR 431040 | Zbl 0291.31002