# American Institute of Mathematical Sciences

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

 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
Citation: Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems - A, 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 (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.  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)., (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.  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, 224 (1983).   Google Scholar [6] Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations,, Comm. Pure Appl. Math., 38 (1985), 679.  doi: 10.1002/cpa.3160380515.  Google Scholar [7] L. E. Payne, Some remarks on maximum principles,, J. Analyse Math., 30 (1976), 421.  doi: 10.1007/BF02786729.  Google Scholar [8] Vladimir E. Shklover, Schiffer problem and isoparametric hypersurfaces,, Rev. Mat. Iberoamericana, 16 (2000), 529.   Google Scholar [9] René P. Sperb, "Maximum Principles and their Applications,", Mathematics in Science and Engineering, 157 (1981).   Google Scholar [10] Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations,, Handbook of Differential Geometry, (2000), 963.   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 (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.  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)., (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.  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, 224 (1983).   Google Scholar [6] Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations,, Comm. Pure Appl. Math., 38 (1985), 679.  doi: 10.1002/cpa.3160380515.  Google Scholar [7] L. E. Payne, Some remarks on maximum principles,, J. Analyse Math., 30 (1976), 421.  doi: 10.1007/BF02786729.  Google Scholar [8] Vladimir E. Shklover, Schiffer problem and isoparametric hypersurfaces,, Rev. Mat. Iberoamericana, 16 (2000), 529.   Google Scholar [9] René P. Sperb, "Maximum Principles and their Applications,", Mathematics in Science and Engineering, 157 (1981).   Google Scholar [10] Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations,, Handbook of Differential Geometry, (2000), 963.   Google Scholar [11] Jiaping Wang, "Lecture Notes on, Geometric Analysis, ().   Google Scholar
 [1] Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 [2] Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 [3] Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225 [4] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [5] Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 [6] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [7] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [8] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

2019 Impact Factor: 1.338