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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 Università 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. 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. 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 {\tt http://www.math.utexas.edu/mp$\_$arc/}, (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. 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. Google Scholar [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. 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. 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. 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, (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, (1990). 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, (1983). Google Scholar [13] Bernd Kawohl, "Symmetrization-or how to Prove Symmetry of Solutions to a PDE,", Partial differential equations (Praha, (1998), 214. Google Scholar [14] Olga A. Ladyzhenskaya and Nina N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", translated from the Russian by Scripta Technica, (1968). Google Scholar [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. 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. 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. doi: 10.1002/cpa.3160380515. Google Scholar [18] Lawrence E. Payne, Some remarks on maximum principles,, J. Analyse Math., 30 (1976), 421. doi: 10.1007/BF02786729. Google Scholar [19] Patrizia Pucci and James Serrin, "The Maximum Principle,", Progress in Nonlinear Differential Equations and their Applications, (2007). Google Scholar [20] Renè P. Sperb, "Maximum Principles and Their Applications,", volume 157 of Mathematics in Science and Engineering. Academic Press Inc, (1981). Google Scholar [21] Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations,, Handbook of differential geometry, (2000), 963. 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. 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. 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 {\tt http://www.math.utexas.edu/mp$\_$arc/}, (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. 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. Google Scholar [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. 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. 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. 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, (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, (1990). 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, (1983). Google Scholar [13] Bernd Kawohl, "Symmetrization-or how to Prove Symmetry of Solutions to a PDE,", Partial differential equations (Praha, (1998), 214. Google Scholar [14] Olga A. Ladyzhenskaya and Nina N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,", translated from the Russian by Scripta Technica, (1968). Google Scholar [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. 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. 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. doi: 10.1002/cpa.3160380515. Google Scholar [18] Lawrence E. Payne, Some remarks on maximum principles,, J. Analyse Math., 30 (1976), 421. doi: 10.1007/BF02786729. Google Scholar [19] Patrizia Pucci and James Serrin, "The Maximum Principle,", Progress in Nonlinear Differential Equations and their Applications, (2007). Google Scholar [20] Renè P. Sperb, "Maximum Principles and Their Applications,", volume 157 of Mathematics in Science and Engineering. Academic Press Inc, (1981). Google Scholar [21] Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations,, Handbook of differential geometry, (2000), 963. Google Scholar
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