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Nondegeneracy of Solution to the Allen-Cahn Equation with Regular Triangle Symmetry

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DOI: 10.4236/apm.2014.44017    3,599 Downloads   5,205 Views  
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ABSTRACT

The Allen-Cahn equation on the plane has a 6-end solution U with regular triangle symmetry. The angle between consecutive nodal lines of U is . We prove in this paper that U is non-degenerated in the class of functions possessing regular triangle symmetry. As an application, we show the existence of a family of solutions close to U.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Liu, Y. and Wang, J. (2014) Nondegeneracy of Solution to the Allen-Cahn Equation with Regular Triangle Symmetry. Advances in Pure Mathematics, 4, 103-107. doi: 10.4236/apm.2014.44017.

References

[1] Ambrosio, L. and Cabre, X. (2000) Entire Solution of Semilinear Elliptic Equations in R3 and a Conjecture of De Giorgi. Journal of the American Mathematical Society, 13, 725-739. http://dx.doi.org/10.1090/S0894-0347-00-00345-3
[2] Del Pino, M. Kowalczyk M. and Wei, J. (2011) On De Giorgi’s Conjecture in Dimension N ≥ 9. Annals of Mathematics, 174, 1485-1569.
[3] Ghoussoub, N. and Gui, C. (1998) On a Conjecture of De Giorgi and Some Related Problems. Mathematische Annalen, 311, 481-491. http://dx.doi.org/10.1007/s002080050196
[4] Ghoussoub, N. and Gui, C. (1998) On a Conjecture of De Giorgi and Some Related Problems. Mathematische Annalen, 311, 121-132. http://dx.doi.org/10.1007/s002080050196
[5] Savin, O. (2010) Phase Transitions, Minimal Surfaces and a Conjecture of De Giorgi. Current Developments in Mathematics, 2009, 59-113.
[6] Alessio, F., Calamai, A. and Montecchiari, P. (2007) Saddle-Type Solutions for a Class of Semilinear Elliptic Equations. Advances in Differential Equations, 12, 361-380.
[7] Kowalczyk, M., Liu, Y. and Pacard, F. (2012) The Space of 4-Ended Solutions to the Allen-Cahn Equation in the Plane. Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire, 29,761-781.
http://dx.doi.org/10.1016/j.anihpc.2012.04.003
[8] Kowalczyk, M., Liu, Y. and Pacard, F. (2013) The Classification of Four-End Solutions to the Allen-Cahn Equation on the Plane. Analysis and PDE, 6, 1675-1718.
[9] Del Pino, M., Kowalczyk, M., Pacard, F. and Wei, J. (2010) Multiple-End Solutions to the Allen-Cahn Equation in R2. Journal of Functional Analysis, 258, 458-503. http://dx.doi.org/10.1016/j.jfa.2009.04.020
[10] Del Pino, M., Kowalczyk, M. and Pacard, F. (2013) Moduli Space Theory for the Allen-Cahn Equation in the Plane. Transactions of the American Mathematical Society, 365, 721-766.
http://dx.doi.org/10.1090/S0002-9947-2012-05594-2
[11] Kowalczyk, M. and Liu, Y. (2011) Nondegeneracy of the Saddle Solution of the Allen-Cahn Equation on the Plane. Proceedings of the American Mathematical Society, 139, 4319-4329.
http://dx.doi.org/10.1090/S0002-9939-2011-11217-6
[12] Gui, C. (2008) Hamiltonian Identities for Elliptic Partial Differential Equations. Journal of Functional Analysis, 254, 904-933. http://dx.doi.org/10.1016/j.jfa.2007.10.015

  
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