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

Diego Castellaneta; Alberto Farina; Enrico Valdinoci

doi:10.3934/cpaa.2012.11.1983

Article Contents

2012, Volume 11, Issue 5: 1983-2003. Doi: 10.3934/cpaa.2012.11.1983

This issue Previous Article Solvability and asymptotic analysis of a generalization of the Caginalp phase field system Next Article Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities

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 scientica, 1, I-00133 Rome
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: May 31, 2011

Revised: October 31, 2011

Published: August 2012

Abstract Related Papers Cited by

Abstract

We consider a singular or degenerate elliptic problem in a proper domain and we prove a gradient bound and some symmetry results.

Keywords:

Singular and degenerate elliptic pde's,
a priori estimates.

Mathematics Subject Classification: Primary: 35J92; Secondary: 5J70, 35J75.

Citation:

Related Papers

Cited by

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.