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September  2012, 11(5): 1983-2003. doi: 10.3934/cpaa.2012.11.1983

A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature

Diego Castellaneta 1, , Alberto Farina 2, and  Enrico Valdinoci 3,

1. 

Università di Roma Tor Vergata, Dipartimento di Matematica, via della ricerca scienti ca, 1, I-00133 Rome, Italy
2. 

LAMFA – CNRS UMR 6140, Université de Picardie Jules Verne, 33, rue Saint-Leu 80039 Amiens CEDEX 1
3. 

Università di Roma Tor Vergata, Dipartimento di Matematica, Via della Ricerca Scientifica, I-00133 Rome

Received  June 2011 Revised  November 2011 Published  March 2012

We consider a singular or degenerate elliptic problem in a proper domain and we prove a gradient bound and some symmetry results.
Keywords: a priori estimates., Singular and degenerate elliptic pde's.
Mathematics Subject Classification: Primary: 35J92; Secondary: 5J70, 35J7.
Citation: Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983
References:
[1]

S. S. Antman, Nonuniqueness of equilibrium states for bars in tension, J. Math. Anal. Appl., 44 (1973), 333-349. doi: 10.1016/0022-247X(73)90063-2.  Google Scholar

[2]

Luis Caffarelli, Nicola Garofalo and Fausto Segala, A gradient bound for entire solutions of quasi-linear equations and its conseguences, Comm. Pure Appl. Math., 47 (1994), 1457-1473. doi: 10.1002/cpa.3160471103.  Google Scholar

[3]

Diego Castellaneta, Stima puntuale del gradiente per soluzioni di equazioni ellittiche singolari o degeneri in domini propri con curvatura media nonnegativa, avaliable online at http://www.math.utexas.edu/mp$\_$arc/, Tesi di laurea specialistica, Università di Roma Tor Vergata, 2009. Google Scholar

[4]

Emmanuele DiBenedetto, "Degerate Parabolic Equation," Springer-Verlag, 1991.  Google Scholar

[5]

James Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57. doi: 10.1007/BF03023575.  Google Scholar

[6]

Alberto Farina, Berardino Sciunzi and Enrico Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 741-791.  Google Scholar

[7]

Alberto Farina and Enrico Valdinoci, Geometry of quasiminimal phase transitions, Calc. Var. Partial Differ. Equ., 33 (2008), 1-35. doi: 10.1007/s00526-007-0146-1.  Google Scholar

[8]

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

[9]

Alberto Farina and Enrico Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. Ration. Mech. Anal., 195 (2010), 1025-1058. doi: 10.1007/s00205-009-0227-8.  Google Scholar

[10]

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, volume 224 of "Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]," Springer-Verlag, Berlin, second edition, 1983.  Google Scholar

[11]

Lars Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Reprint of the second (1990) edition. Classics in Mathematics. Springer-Verlag, Berlin, 2003.  Google Scholar

[12]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients," Reprint of the 1983 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005.  Google Scholar

[13]

Bernd Kawohl, "Symmetrization-or how to Prove Symmetry of Solutions to a PDE," Partial differential equations (Praha, 1998), 214-229, Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, 2000.  Google Scholar

[14]

Olga A. Ladyzhenskaya and Nina N. Ural'tseva, "Linear and Quasilinear Elliptic Equations," translated from the Russian by Scripta Technica, Academic Press, New York-London, 1968.  Google Scholar

[15]

Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[16]

Rafael de la Llave and Enrico Valdinoci, Ground states and critical points for generalized Frankel-Kontorova models in $Z^d$, Nonlinearity, 20 (2007), 2409-2424. doi: 10.1088/0951-7715/20/10/008.  Google Scholar

[17]

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

[18]

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

[19]

Patrizia Pucci and James Serrin, "The Maximum Principle," Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[20]

Renè P. Sperb, "Maximum Principles and Their Applications," volume 157 of Mathematics in Science and Engineering. Academic Press Inc, [Harcourt Brace Jovanovich Publishers], New York, 1981.  Google Scholar

[21]

Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of differential geometry, Vol. I, 963-995, North-Holland, Amsterdam, 2000.  Google Scholar

show all references

References:
[1]

S. S. Antman, Nonuniqueness of equilibrium states for bars in tension, J. Math. Anal. Appl., 44 (1973), 333-349. doi: 10.1016/0022-247X(73)90063-2.  Google Scholar

[2]

Luis Caffarelli, Nicola Garofalo and Fausto Segala, A gradient bound for entire solutions of quasi-linear equations and its conseguences, Comm. Pure Appl. Math., 47 (1994), 1457-1473. doi: 10.1002/cpa.3160471103.  Google Scholar

[3]

Diego Castellaneta, Stima puntuale del gradiente per soluzioni di equazioni ellittiche singolari o degeneri in domini propri con curvatura media nonnegativa, avaliable online at http://www.math.utexas.edu/mp$\_$arc/, Tesi di laurea specialistica, Università di Roma Tor Vergata, 2009. Google Scholar

[4]

Emmanuele DiBenedetto, "Degerate Parabolic Equation," Springer-Verlag, 1991.  Google Scholar

[5]

James Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57. doi: 10.1007/BF03023575.  Google Scholar

[6]

Alberto Farina, Berardino Sciunzi and Enrico Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 741-791.  Google Scholar

[7]

Alberto Farina and Enrico Valdinoci, Geometry of quasiminimal phase transitions, Calc. Var. Partial Differ. Equ., 33 (2008), 1-35. doi: 10.1007/s00526-007-0146-1.  Google Scholar

[8]

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

[9]

Alberto Farina and Enrico Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. Ration. Mech. Anal., 195 (2010), 1025-1058. doi: 10.1007/s00205-009-0227-8.  Google Scholar

[10]

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, volume 224 of "Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]," Springer-Verlag, Berlin, second edition, 1983.  Google Scholar

[11]

Lars Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Reprint of the second (1990) edition. Classics in Mathematics. Springer-Verlag, Berlin, 2003.  Google Scholar

[12]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients," Reprint of the 1983 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005.  Google Scholar

[13]

Bernd Kawohl, "Symmetrization-or how to Prove Symmetry of Solutions to a PDE," Partial differential equations (Praha, 1998), 214-229, Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, 2000.  Google Scholar

[14]

Olga A. Ladyzhenskaya and Nina N. Ural'tseva, "Linear and Quasilinear Elliptic Equations," translated from the Russian by Scripta Technica, Academic Press, New York-London, 1968.  Google Scholar

[15]

Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[16]

Rafael de la Llave and Enrico Valdinoci, Ground states and critical points for generalized Frankel-Kontorova models in $Z^d$, Nonlinearity, 20 (2007), 2409-2424. doi: 10.1088/0951-7715/20/10/008.  Google Scholar

[17]

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

[18]

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

[19]

Patrizia Pucci and James Serrin, "The Maximum Principle," Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[20]

Renè P. Sperb, "Maximum Principles and Their Applications," volume 157 of Mathematics in Science and Engineering. Academic Press Inc, [Harcourt Brace Jovanovich Publishers], New York, 1981.  Google Scholar

[21]

Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of differential geometry, Vol. I, 963-995, North-Holland, Amsterdam, 2000.  Google Scholar

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