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.
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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