A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature

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November 2011, 30(4): 1139-1144. doi: 10.3934/dcds.2011.30.1139

A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature

Alberto Farina ^1, and Enrico Valdinoci ^2,

1.	LAMFA – CNRS UMR 6140, Université de Picardie Jules Verne, 33, rue Saint-Leu 80039 Amiens CEDEX 1, France
2.	Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano

Received February 2010 Revised August 2010 Published May 2011

Abstract
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Keywords: a-priori estimates., Geometric analysis of PDEs.

Mathematics Subject Classification: Primary: 35J15, 35J61; Secondary: 53B2.

Citation: Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139

References:

[1]	Marcel Berger, Paul Gauduchon and Edmond Mazet, "Le Spectre d'une Variété Riemannienne," Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin, 1971.
[2]	Luis Caffarelli, Nicola Garofalo and Fausto Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math., 47 (1994), 1457-1473. doi: 10.1002/cpa.3160471103.
[3]	Alberto Farina, Yannick Sire and Enrico Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds, preprint (2008).
[4]	Alberto Farina and Enrico Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature, Adv. Math., 225 (2010), 2808-2827. doi: 10.1016/j.aim.2010.05.008.
[5]	David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences], 2^nd edition, 224, Springer-Verlag, Berlin, 1983.
[6]	Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684. doi: 10.1002/cpa.3160380515.
[7]	L. E. Payne, Some remarks on maximum principles, J. Analyse Math., 30 (1976), 421-433. doi: 10.1007/BF02786729.
[8]	Vladimir E. Shklover, Schiffer problem and isoparametric hypersurfaces, Rev. Mat. Iberoamericana, 16 (2000), 529-569.
[9]	René P. Sperb, "Maximum Principles and their Applications," Mathematics in Science and Engineering, 157, Academic Press Inc., New York, 1981.
[10]	Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of Differential Geometry, I, North-Holland, Amsterdam, (2000), 963-995.
[11]	Jiaping Wang, "Lecture Notes on Geometric Analysis".

show all references

References:

[1]	Marcel Berger, Paul Gauduchon and Edmond Mazet, "Le Spectre d'une Variété Riemannienne," Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin, 1971.
[2]	Luis Caffarelli, Nicola Garofalo and Fausto Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math., 47 (1994), 1457-1473. doi: 10.1002/cpa.3160471103.
[3]	Alberto Farina, Yannick Sire and Enrico Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds, preprint (2008).
[4]	Alberto Farina and Enrico Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature, Adv. Math., 225 (2010), 2808-2827. doi: 10.1016/j.aim.2010.05.008.
[5]	David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences], 2^nd edition, 224, Springer-Verlag, Berlin, 1983.
[6]	Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684. doi: 10.1002/cpa.3160380515.
[7]	L. E. Payne, Some remarks on maximum principles, J. Analyse Math., 30 (1976), 421-433. doi: 10.1007/BF02786729.
[8]	Vladimir E. Shklover, Schiffer problem and isoparametric hypersurfaces, Rev. Mat. Iberoamericana, 16 (2000), 529-569.
[9]	René P. Sperb, "Maximum Principles and their Applications," Mathematics in Science and Engineering, 157, Academic Press Inc., New York, 1981.
[10]	Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of Differential Geometry, I, North-Holland, Amsterdam, (2000), 963-995.
[11]	Jiaping Wang, "Lecture Notes on Geometric Analysis".

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