One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane

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May 2012, 11(3): 1157-1166. doi: 10.3934/cpaa.2012.11.1157

One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane

Giuseppe Riey ^1, and Berardino Sciunzi ^2,

1.	Dipartimento di Matematica, Università della Calabria, Ponte P. Bucci 31B, 87036 Arcavacata di Rende, Cosenza, Italy
2.	Dipartimento di Matematica, Universitá della Calabria, V. P. Bucci, I-87036 Arcavacata di Rende (CS)

Received December 2010 Revised March 2011 Published December 2011

Abstract
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We prove one-dimensional symmetry of monotone solutions for some anisotropic quasilinear elliptic equations in the plane.

Keywords: geometric analysis, Degenerate anisotropic elliptic PDEs, rigidity and symmetry results..

Mathematics Subject Classification: 35J70, 35J60, 35B05, 35B09, 35B5.

Citation: Giuseppe Riey, Berardino Sciunzi. One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1157-1166. doi: 10.3934/cpaa.2012.11.1157

References:

[1]	G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property,, Acta Appl. Math., 65 (2001), 9. doi: 10.1023/A:1010602715526.
[2]	L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $R^3$ and a conjecture of De Giorgi,, J. Amer. Math. Soc., 13 (2000), 725. doi: 10.1090/S0894-0347-00-00345-3.
[3]	I. Birindelli and E. Valdinoci, The Ginzburg-Landau equation in the Heisenberg group,, Commun. Contemp. Math., 10 (2008), 671. doi: 10.1142/S0219199708002946.
[4]	L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations,, J. Differential Equations, 206 (2004), 483. doi: 10.1016/j.jde.2004.05.012.
[5]	E. De Giorgi, Convergence problems for functionals and operators,, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1979), 131.
[6]	M. Del Pino, M. Kowalczyk and J. Wei, A counterexample to a conjecture by De Giorgi in large dimensions,, C. R. Math. Acad. Sci. Paris, 346 (2008), 1261. doi: 10.1016/j.crma.2008.10.010.
[7]	D. De Silva and O. Savin, Symmetry of global solutions to a class of fully nonlinear elliptic equations in $2$D,, Indiana Univ. Math. J., 58 (2009), 301. doi: 10.1512/iumj.2009.58.3396.
[8]	A. Farina, Symmetry for solutions of semilinear elliptic equations in $R^N$ and related conjectures,, Ricerche Mat., (1999), 129. doi: 2001h:35056.
[9]	A. Farina, Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires,, Habilitation \`a diriger des recherches, (2002).
[10]	A. Farina, One-dimensional symmetry for solutions of quasilinear equations in $R^2$,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 6 (2003), 685. doi: 2005h:35093.
[11]	A. Farina, Liouville-type theorems for elliptic problems, in, Elsevier North-Holland, Amsterdam, (2003), 61-116., (2003), 61. doi: 10.1016/S1874-5733(07)80005-2.
[12]	A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach,, Ann. Sc. Norm. Super. Pisa, 7 (2008), 741. doi: 2009j:58020.
[13]	A. Farina, B. Sciunzi and E. Valdinoci, On a Poincaré type formula for solutions of singular and degenerate elliptic equations,, Manuscripta Math., 132 (2010), 335. doi: 10.1007/s00229-010-0349-1.
[14]	A. Farina, Y. Sire and E. Valdinoci, Stable solutions of elliptic equations on riemannian manifolds,, Preprint, (2008).
[15]	A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions,, Calc. Var. Partial Differential Equations, 33 (2008), 1. doi: 10.1007/s00526-007-0146-1.
[16]	A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems,, Ser. Adv. Math. Appl. Sci. World Sci. Publ., (2008). doi: 10.1142/9789812834744_0004.
[17]	A. Farina and E. Valdinoci, 1D symmetry for solutions of semilinear and quasilinear elliptic equations,, Trans. Amer Math. Soc., 363 (2011), 579. doi: 10.1090/S0002-9947-2010-05021-4.
[18]	F. Ferrari and E. Valdinoci, A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems,, Math. Ann., 343 (2009), 351. doi: 10.1007/s00208-008-0274-8.
[19]	I. Fragalà, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 21 (2004), 715. doi: 10.1016/j.anihpc.2003.12.001.
[20]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1997). doi: 2001k:35004.
[21]	N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems,, Math. Ann., 311 (1998), 481. doi: 10.1007/s002080050196.
[22]	P. Le and B. Sciunzi, Regularity of solutions of degenerate quasilinear elliptic equations,, In preparation., ().
[23]	N. G. Meyers and J. Serrin, $H=W$,, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055.
[24]	M. Mihâilescu, P. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent,, J. Math. Anal. Appl., 340 (2008), 687. doi: 10.1016/j.jmaa.2007.09.015.
[25]	J. Rácosník, Some remarks to anisotropic Sobolev spaces I,, Beitrage Anal., 13 (1979), 55.
[26]	J. Rácosník, Some remarks to anisotropic Sobolev spaces II,, Beitrage Anal., 15 (1981), 127.
[27]	O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. doi: 2009m:58025.
[28]	B. Sciunzi and E. Valdinoci, Mean curvature properties for $p$-Laplace phase transitions,, J. Eur. Math. Soc. (JEMS), 7 (2005), 319. doi: 006m:35055.
[29]	E. Valdinoci, B. Sciunzi and O. Savin, Flat level set regularity of $p$-Laplace phase transitions,, Mem. Amer. Math. Soc., 182 (2006). doi: 10.4007/annals.2009.169.41.
[30]	Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result,, J. Funct. Anal., 256 (2009), 1842. doi: 10.1016/j.jfa.2009.01.020.
[31]	P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational. Mech. Anal., 141 (1998), 375. doi: 99c:49045.
[32]	M. Troisi, Teoremi di inclusioni per spazi di Sobolev non isotropi,, Ricerche Mat., 18 (1969), 3.
[33]	L. Ven'tuan, On embedding theorems for spaces of functions with partial derivatives of various degree of summability,, Vestnik Leingrad. Univ., 16 (1961), 23.

show all references

References:

[1]	G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property,, Acta Appl. Math., 65 (2001), 9. doi: 10.1023/A:1010602715526.
[2]	L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $R^3$ and a conjecture of De Giorgi,, J. Amer. Math. Soc., 13 (2000), 725. doi: 10.1090/S0894-0347-00-00345-3.
[3]	I. Birindelli and E. Valdinoci, The Ginzburg-Landau equation in the Heisenberg group,, Commun. Contemp. Math., 10 (2008), 671. doi: 10.1142/S0219199708002946.
[4]	L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations,, J. Differential Equations, 206 (2004), 483. doi: 10.1016/j.jde.2004.05.012.
[5]	E. De Giorgi, Convergence problems for functionals and operators,, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1979), 131.
[6]	M. Del Pino, M. Kowalczyk and J. Wei, A counterexample to a conjecture by De Giorgi in large dimensions,, C. R. Math. Acad. Sci. Paris, 346 (2008), 1261. doi: 10.1016/j.crma.2008.10.010.
[7]	D. De Silva and O. Savin, Symmetry of global solutions to a class of fully nonlinear elliptic equations in $2$D,, Indiana Univ. Math. J., 58 (2009), 301. doi: 10.1512/iumj.2009.58.3396.
[8]	A. Farina, Symmetry for solutions of semilinear elliptic equations in $R^N$ and related conjectures,, Ricerche Mat., (1999), 129. doi: 2001h:35056.
[9]	A. Farina, Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires,, Habilitation \`a diriger des recherches, (2002).
[10]	A. Farina, One-dimensional symmetry for solutions of quasilinear equations in $R^2$,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 6 (2003), 685. doi: 2005h:35093.
[11]	A. Farina, Liouville-type theorems for elliptic problems, in, Elsevier North-Holland, Amsterdam, (2003), 61-116., (2003), 61. doi: 10.1016/S1874-5733(07)80005-2.
[12]	A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach,, Ann. Sc. Norm. Super. Pisa, 7 (2008), 741. doi: 2009j:58020.
[13]	A. Farina, B. Sciunzi and E. Valdinoci, On a Poincaré type formula for solutions of singular and degenerate elliptic equations,, Manuscripta Math., 132 (2010), 335. doi: 10.1007/s00229-010-0349-1.
[14]	A. Farina, Y. Sire and E. Valdinoci, Stable solutions of elliptic equations on riemannian manifolds,, Preprint, (2008).
[15]	A. Farina and E. Valdinoci, Geometry of quasiminimal phase transitions,, Calc. Var. Partial Differential Equations, 33 (2008), 1. doi: 10.1007/s00526-007-0146-1.
[16]	A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems,, Ser. Adv. Math. Appl. Sci. World Sci. Publ., (2008). doi: 10.1142/9789812834744_0004.
[17]	A. Farina and E. Valdinoci, 1D symmetry for solutions of semilinear and quasilinear elliptic equations,, Trans. Amer Math. Soc., 363 (2011), 579. doi: 10.1090/S0002-9947-2010-05021-4.
[18]	F. Ferrari and E. Valdinoci, A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems,, Math. Ann., 343 (2009), 351. doi: 10.1007/s00208-008-0274-8.
[19]	I. Fragalà, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 21 (2004), 715. doi: 10.1016/j.anihpc.2003.12.001.
[20]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1997). doi: 2001k:35004.
[21]	N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems,, Math. Ann., 311 (1998), 481. doi: 10.1007/s002080050196.
[22]	P. Le and B. Sciunzi, Regularity of solutions of degenerate quasilinear elliptic equations,, In preparation., ().
[23]	N. G. Meyers and J. Serrin, $H=W$,, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055.
[24]	M. Mihâilescu, P. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent,, J. Math. Anal. Appl., 340 (2008), 687. doi: 10.1016/j.jmaa.2007.09.015.
[25]	J. Rácosník, Some remarks to anisotropic Sobolev spaces I,, Beitrage Anal., 13 (1979), 55.
[26]	J. Rácosník, Some remarks to anisotropic Sobolev spaces II,, Beitrage Anal., 15 (1981), 127.
[27]	O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41. doi: 2009m:58025.
[28]	B. Sciunzi and E. Valdinoci, Mean curvature properties for $p$-Laplace phase transitions,, J. Eur. Math. Soc. (JEMS), 7 (2005), 319. doi: 006m:35055.
[29]	E. Valdinoci, B. Sciunzi and O. Savin, Flat level set regularity of $p$-Laplace phase transitions,, Mem. Amer. Math. Soc., 182 (2006). doi: 10.4007/annals.2009.169.41.
[30]	Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result,, J. Funct. Anal., 256 (2009), 1842. doi: 10.1016/j.jfa.2009.01.020.
[31]	P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational. Mech. Anal., 141 (1998), 375. doi: 99c:49045.
[32]	M. Troisi, Teoremi di inclusioni per spazi di Sobolev non isotropi,, Ricerche Mat., 18 (1969), 3.
[33]	L. Ven'tuan, On embedding theorems for spaces of functions with partial derivatives of various degree of summability,, Vestnik Leingrad. Univ., 16 (1961), 23.

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