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 Δu-F ' (u)=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 θ(0,π/2), 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, π-θ, π+θ and 2π-θ 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 k2.

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/

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