The space of 4-ended solutions to the Allen–Cahn equation in the plane
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 5, pp. 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 : 10.1016/j.anihpc.2012.04.003
Mots clés : Allen–Cahn equation, Classification of solutions, Entire solutions of semilinear elliptic equations
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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, Tome 29 (2012) no. 5, pp. 761-781. doi : 10.1016/j.anihpc.2012.04.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.04.003/

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