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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

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.  Google Scholar

[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.  Google Scholar

[3]

I. Birindelli and E. Valdinoci, The Ginzburg-Landau equation in the Heisenberg group,, Commun. Contemp. Math., 10 (2008), 671.  doi: 10.1142/S0219199708002946.  Google Scholar

[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.  Google Scholar

[5]

E. De Giorgi, Convergence problems for functionals and operators,, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1979), 131.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Farina, Symmetry for solutions of semilinear elliptic equations in $R^N$ and related conjectures,, Ricerche Mat., (1999), 129.  doi: 2001h:35056.  Google Scholar

[9]

A. Farina, Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires,, Habilitation \`a diriger des recherches, (2002).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Farina, Y. Sire and E. Valdinoci, Stable solutions of elliptic equations on riemannian manifolds,, Preprint, (2008).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1997).  doi: 2001k:35004.  Google Scholar

[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.  Google Scholar

[22]

P. Le and B. Sciunzi, Regularity of solutions of degenerate quasilinear elliptic equations,, In preparation., ().   Google Scholar

[23]

N. G. Meyers and J. Serrin, $H=W$,, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055.   Google Scholar

[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.  Google Scholar

[25]

J. Rácosník, Some remarks to anisotropic Sobolev spaces I,, Beitrage Anal., 13 (1979), 55.   Google Scholar

[26]

J. Rácosník, Some remarks to anisotropic Sobolev spaces II,, Beitrage Anal., 15 (1981), 127.   Google Scholar

[27]

O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41.  doi: 2009m:58025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational. Mech. Anal., 141 (1998), 375.  doi: 99c:49045.  Google Scholar

[32]

M. Troisi, Teoremi di inclusioni per spazi di Sobolev non isotropi,, Ricerche Mat., 18 (1969), 3.   Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

I. Birindelli and E. Valdinoci, The Ginzburg-Landau equation in the Heisenberg group,, Commun. Contemp. Math., 10 (2008), 671.  doi: 10.1142/S0219199708002946.  Google Scholar

[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.  Google Scholar

[5]

E. De Giorgi, Convergence problems for functionals and operators,, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1979), 131.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Farina, Symmetry for solutions of semilinear elliptic equations in $R^N$ and related conjectures,, Ricerche Mat., (1999), 129.  doi: 2001h:35056.  Google Scholar

[9]

A. Farina, Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires,, Habilitation \`a diriger des recherches, (2002).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Farina, Y. Sire and E. Valdinoci, Stable solutions of elliptic equations on riemannian manifolds,, Preprint, (2008).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer, (1997).  doi: 2001k:35004.  Google Scholar

[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.  Google Scholar

[22]

P. Le and B. Sciunzi, Regularity of solutions of degenerate quasilinear elliptic equations,, In preparation., ().   Google Scholar

[23]

N. G. Meyers and J. Serrin, $H=W$,, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055.   Google Scholar

[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.  Google Scholar

[25]

J. Rácosník, Some remarks to anisotropic Sobolev spaces I,, Beitrage Anal., 13 (1979), 55.   Google Scholar

[26]

J. Rácosník, Some remarks to anisotropic Sobolev spaces II,, Beitrage Anal., 15 (1981), 127.   Google Scholar

[27]

O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math., 169 (2009), 41.  doi: 2009m:58025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational. Mech. Anal., 141 (1998), 375.  doi: 99c:49045.  Google Scholar

[32]

M. Troisi, Teoremi di inclusioni per spazi di Sobolev non isotropi,, Ricerche Mat., 18 (1969), 3.   Google Scholar

[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.   Google Scholar

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