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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

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.
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. 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. 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 {\tt http://www.math.utexas.edu/mp$\_$arc/}, (2009).

[4]

Emmanuele DiBenedetto, "Degerate Parabolic Equation,", Springer-Verlag, (1991).

[5]

James Eells, The surfaces of Delaunay,, Math. Intelligencer, 9 (1987), 53. 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.

[7]

Alberto Farina and Enrico Valdinoci, Geometry of quasiminimal phase transitions,, Calc. Var. Partial Differ. Equ., 33 (2008), 1. 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. 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. doi: 10.1007/s00205-009-0227-8.

[10]

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order,, volume 224 of, (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, (1990).

[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, (1983).

[13]

Bernd Kawohl, "Symmetrization-or how to Prove Symmetry of Solutions to a PDE,", Partial differential equations (Praha, (1998), 214.

[14]

Olga A. Ladyzhenskaya and Nina N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", translated from the Russian by Scripta Technica, (1968).

[15]

Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203. 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. 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. doi: 10.1002/cpa.3160380515.

[18]

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

[19]

Patrizia Pucci and James Serrin, "The Maximum Principle,", Progress in Nonlinear Differential Equations and their Applications, (2007).

[20]

Renè P. Sperb, "Maximum Principles and Their Applications,", volume 157 of Mathematics in Science and Engineering. Academic Press Inc, (1981).

[21]

Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations,, Handbook of differential geometry, (2000), 963.

show all references

References:
[1]

S. S. Antman, Nonuniqueness of equilibrium states for bars in tension,, J. Math. Anal. Appl., 44 (1973), 333. 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. 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 {\tt http://www.math.utexas.edu/mp$\_$arc/}, (2009).

[4]

Emmanuele DiBenedetto, "Degerate Parabolic Equation,", Springer-Verlag, (1991).

[5]

James Eells, The surfaces of Delaunay,, Math. Intelligencer, 9 (1987), 53. 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.

[7]

Alberto Farina and Enrico Valdinoci, Geometry of quasiminimal phase transitions,, Calc. Var. Partial Differ. Equ., 33 (2008), 1. 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. 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. doi: 10.1007/s00205-009-0227-8.

[10]

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order,, volume 224 of, (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, (1990).

[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, (1983).

[13]

Bernd Kawohl, "Symmetrization-or how to Prove Symmetry of Solutions to a PDE,", Partial differential equations (Praha, (1998), 214.

[14]

Olga A. Ladyzhenskaya and Nina N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", translated from the Russian by Scripta Technica, (1968).

[15]

Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203. 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. 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. doi: 10.1002/cpa.3160380515.

[18]

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

[19]

Patrizia Pucci and James Serrin, "The Maximum Principle,", Progress in Nonlinear Differential Equations and their Applications, (2007).

[20]

Renè P. Sperb, "Maximum Principles and Their Applications,", volume 157 of Mathematics in Science and Engineering. Academic Press Inc, (1981).

[21]

Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations,, Handbook of differential geometry, (2000), 963.

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