The space of 4-ended solutions to the Allen–Cahn equation in the plane
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 5, p. 761-781

We are interested in entire solutions of the Allen–Cahn equation $\Delta u-{F}^{\text{'}}\left(u\right)=0$ which have some special structure at infinity. In this equation, the function F is an even, double well potential. The solutions we are interested in have their zero set asymptotic to 4 half oriented affine lines at infinity and, along each of these half affine lines, the solutions are asymptotic to the one dimensional heteroclinic solution: such solutions are called 4-ended solutions. The main result of our paper states that, for any $\theta \in \left(0,\pi /2\right)$, there exists a 4-ended solution of the Allen–Cahn equation whose zero set is at infinity asymptotic to the half oriented affine lines making the angles θ, $\pi -\theta$, $\pi +\theta$ and $2\pi -\theta$ with the x-axis. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen–Cahn equation in dimension 2, for $k⩾2$.

DOI : https://doi.org/10.1016/j.anihpc.2012.04.003
Keywords: Allen–Cahn equation, Classification of solutions, Entire solutions of semilinear elliptic equations
@article{AIHPC_2012__29_5_761_0,
author = {Kowalczyk, Micha\l\ and Liu, Yong and Pacard, Frank},
title = {The space of 4-ended solutions to the Allen--Cahn equation in the plane},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {29},
number = {5},
year = {2012},
pages = {761-781},
doi = {10.1016/j.anihpc.2012.04.003},
zbl = {1254.35219},
mrnumber = {2971030},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2012__29_5_761_0}
}

Kowalczyk, Michał; Liu, Yong; Pacard, Frank. The space of 4-ended solutions to the Allen–Cahn equation in the plane. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 5, pp. 761-781. doi : 10.1016/j.anihpc.2012.04.003. http://www.numdam.org/item/AIHPC_2012__29_5_761_0/

[1] L. Ambrosio, X. Cabré, Entire solutions of semilinear elliptic equations in ${R}^{3}$ and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 no. 4 (2000), 725-739 | MR 1775735 | Zbl 0968.35041

[2] H. Berestycki, L.A. Caffarelli, L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math. 50 no. 11 (1997), 1089-1111 | MR 1470317 | Zbl 0906.35035

[3] X. Cabré, Uniqueness and stability of saddle-shaped solutions to the Allen–Cahn equation, arXiv e-prints, 2011. | MR 2960334

[4] H. Dang, P.C. Fife, L.A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys. 43 no. 6 (1992), 984-998 | MR 1198672 | Zbl 0764.35048

[5] M. del Pino, M. Kowalczyk, F. Pacard, Moduli space theory for the Allen–Cahn equation in the plane, Trans. Amer. Math. Soc. (2010), in press. | MR 2995371

[6] M. Del Pino, M. Kowalczyk, F. Pacard, J. Wei, Multiple-end solutions to the Allen–Cahn equation in ${R}^{2}$, J. Funct. Anal. 258 no. 2 (2010), 458-503 | MR 2557944 | Zbl 1203.35108

[7] M. Del Pino, M. Kowalczyk, J. Wei, On De Giorgiʼs conjecture in dimension $N⩾9$, Ann. of Math. (2) 174 no. 3 (2011), 1485-1569 | MR 2846486 | Zbl 1238.35019

[8] B. Devyver, On the finiteness of the Morse index for Schrödinger operators, arXiv:1011.3390v2 (2011) | MR 2959680

[9] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 no. 1 (1985), 121-132 | MR 808112 | Zbl 0573.53038

[10] N. Ghoussoub, C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 no. 3 (1998), 481-491 | MR 1637919 | Zbl 0918.35046

[11] C. Gui, Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal. 254 no. 4 (2008), 904-933 | MR 2381198 | Zbl 1148.35023

[12] C. Gui, Even symmetry of some entire solutions to the Allen–Cahn equation in two dimensions, arXiv e-prints, 2011. | MR 2911416

[13] H. Karcher, Embedded minimal surfaces derived from Scherkʼs examples, Manuscripta Math. 62 no. 1 (1988), 83-114 | MR 958255 | Zbl 0658.53006

[14] M. Kowalczyk, Y. Liu, Nondegeneracy of the saddle solution of the Allen–Cahn equation on the plane, Proc. Amer. Math. Soc. 139 no. 12 (2011), 4319-4329 | MR 2823077 | Zbl 1241.35079

[15] M. Kowalczyk, Y. Liu, F. Pacard, The classification of four ended solutions to the Allen–Cahn equation on the plane, in preparation, 2011.

[16] J. Perez, M. Traizet, The classification of singly periodic minimal surfaces with genus zero and Scherk type ends, Trans. Amer. Math. Soc. 359 (2007), 965-990 | MR 2262839 | Zbl 1110.53008

[17] O. Savin, Regularity of at level sets in phase transitions, Ann. of Math. (2) 169 no. 1 (2009), 41-78 | MR 2480601 | Zbl 1180.35499

[18] M. Schatzman, On the stability of the saddle solution of Allen–Cahnʼs equation, Proc. Roy. Soc. Edinburgh Sect. A 125 no. 6 (1995), 1241-1275 | MR 1363002 | Zbl 0852.35020