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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

N/A
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 & 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.  Google Scholar

[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.  Google Scholar

[3]

Alberto Farina, Yannick Sire and Enrico Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds, preprint (2008). Google Scholar

[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.  Google Scholar

[5]

David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences], 2nd edition, 224, Springer-Verlag, Berlin, 1983.  Google Scholar

[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.  Google Scholar

[7]

L. E. Payne, Some remarks on maximum principles, J. Analyse Math., 30 (1976), 421-433. doi: 10.1007/BF02786729.  Google Scholar

[8]

Vladimir E. Shklover, Schiffer problem and isoparametric hypersurfaces, Rev. Mat. Iberoamericana, 16 (2000), 529-569.  Google Scholar

[9]

René P. Sperb, "Maximum Principles and their Applications," Mathematics in Science and Engineering, 157, Academic Press Inc., New York, 1981.  Google Scholar

[10]

Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of Differential Geometry, I, North-Holland, Amsterdam, (2000), 963-995.  Google Scholar

[11]

Jiaping Wang, "Lecture Notes on, Geometric Analysis, ().   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

Alberto Farina, Yannick Sire and Enrico Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds, preprint (2008). Google Scholar

[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.  Google Scholar

[5]

David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, [Fundamental Principles of Mathematical Sciences], 2nd edition, 224, Springer-Verlag, Berlin, 1983.  Google Scholar

[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.  Google Scholar

[7]

L. E. Payne, Some remarks on maximum principles, J. Analyse Math., 30 (1976), 421-433. doi: 10.1007/BF02786729.  Google Scholar

[8]

Vladimir E. Shklover, Schiffer problem and isoparametric hypersurfaces, Rev. Mat. Iberoamericana, 16 (2000), 529-569.  Google Scholar

[9]

René P. Sperb, "Maximum Principles and their Applications," Mathematics in Science and Engineering, 157, Academic Press Inc., New York, 1981.  Google Scholar

[10]

Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of Differential Geometry, I, North-Holland, Amsterdam, (2000), 963-995.  Google Scholar

[11]

Jiaping Wang, "Lecture Notes on, Geometric Analysis, ().   Google Scholar

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