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Solvability and asymptotic analysis of a generalization of the Caginalp phase field system
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, 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 |
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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