American Institute of Mathematical Sciences

Advanced
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

[1]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
[2]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258
[3]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[4]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128
[5]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[6]

Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721
[7]

Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277
[8]

Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145
[9]

Laura Baldelli, Roberta Filippucci. A priori estimates for elliptic problems via Liouville type theorems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1883-1898. doi: 10.3934/dcdss.2020148
[10]

Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
[11]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013
[12]

Radjesvarane Alexandre, Jie Liao, Chunjin Lin. Some a priori estimates for the homogeneous Landau equation with soft potentials. Kinetic & Related Models, 2015, 8 (4) : 617-650. doi: 10.3934/krm.2015.8.617
[13]

Monique Chyba, Thomas Haberkorn, Ryan N. Smith, George Wilkens. A geometric analysis of trajectory design for underwater vehicles. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 233-262. doi: 10.3934/dcdsb.2009.11.233
[14]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783
[15]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[16]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731
[17]

Alfonso Castro, Rosa Pardo. A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 783-790. doi: 10.3934/dcdsb.2017038
[18]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025
[19]

Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure & Applied Analysis, 2022, 21 (1) : 275-292. doi: 10.3934/cpaa.2021177
[20]

Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy