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

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

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