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June  2014, 34(6): 2505-2511. doi: 10.3934/dcds.2014.34.2505

Some symmetry results for entire solutions of an elliptic system arising in phase separation

Alberto Farina 1,

1. 

LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens, France

Received  February 2013 Revised  April 2013 Published  December 2013

We study the one dimensional symmetry of entire solutions to an elliptic system arising in phase separation for Bose-Einstein condensates with multiple states. We prove that any monotone solution, with arbitrary algebraic growth at infinity, must be one dimensional in the case of two spatial variables. We also prove the one dimensional symmetry for half-monotone solutions, i.e., for solutions having only one monotone component.
Keywords: Almgrens monotonicity formulae., Symmetry results for elliptic systems, entire solutions.
Mathematics Subject Classification: Primary: 35J4.
Citation: Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505
References:
[1]

H. Berestycki, T.-C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties, Arch. Ration. Mech. Anal., 208 (2013), 163-200. doi: 10.1007/s00205-012-0595-3.

[2]

H. Berestycki, S. Terracini, K. Wang and J. Wei, On entire solutions of an elliptic system modeling phase separation, Adv. Math., 243 (2013), 102-126. doi: 10.1016/j.aim.2013.04.012.

[3]

E. De Giorgi, Convergence problems for functionals and operators, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, 1979, 131-188.

[4]

A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific Publishers, Hackensack, NJ, 2009, 74-96. doi: 10.1142/9789812834744_0004.

[5]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Holder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.

show all references

References:
[1]

H. Berestycki, T.-C. Lin, J. Wei and C. Zhao, On phase-separation model: Asymptotics and qualitative properties, Arch. Ration. Mech. Anal., 208 (2013), 163-200. doi: 10.1007/s00205-012-0595-3.

[2]

H. Berestycki, S. Terracini, K. Wang and J. Wei, On entire solutions of an elliptic system modeling phase separation, Adv. Math., 243 (2013), 102-126. doi: 10.1016/j.aim.2013.04.012.

[3]

E. De Giorgi, Convergence problems for functionals and operators, in Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, 1979, 131-188.

[4]

A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific Publishers, Hackensack, NJ, 2009, 74-96. doi: 10.1142/9789812834744_0004.

[5]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Holder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302.

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