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

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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 scientica, 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

Abstract
Related Papers

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 and 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.
[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.
[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.
[4]	Emmanuele DiBenedetto, "Degerate Parabolic Equation," Springer-Verlag, 1991.
[5]	James Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57. doi: 10.1007/BF03023575.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	Lawrence E. Payne, Some remarks on maximum principles, J. Analyse Math., 30 (1976), 421-433. doi: 10.1007/BF02786729.
[19]	Patrizia Pucci and James Serrin, "The Maximum Principle," Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.
[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.
[21]	Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of differential geometry, Vol. I, 963-995, North-Holland, Amsterdam, 2000.

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.
[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.
[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.
[4]	Emmanuele DiBenedetto, "Degerate Parabolic Equation," Springer-Verlag, 1991.
[5]	James Eells, The surfaces of Delaunay, Math. Intelligencer, 9 (1987), 53-57. doi: 10.1007/BF03023575.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	Lawrence E. Payne, Some remarks on maximum principles, J. Analyse Math., 30 (1976), 421-433. doi: 10.1007/BF02786729.
[19]	Patrizia Pucci and James Serrin, "The Maximum Principle," Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.
[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.
[21]	Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, Handbook of differential geometry, Vol. I, 963-995, North-Holland, Amsterdam, 2000.

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