The space of 4-ended solutions to the Allen-Cahn equation in the plane.

Michal Kowalczyk (1) , Yong Liu (1) , Frank Pacard (2)

1 DIM - Depto Ingeniería Matemática
2 CMLS - Centre de Mathématiques Laurent Schwartz

We are interested in entire solutions of the Allen-Cahn equation Delta u - 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 theta is an element of (0, pi/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 theta, 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.

Domaines: Mathématiques [math]

Michal Kowalczyk, Yong Liu, Frank Pacard. The space of 4-ended solutions to the Allen-Cahn equation in the plane.. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2012, 29 (5), pp.761-781. ⟨10.1016/j.anihpc.2012.04.003⟩. ⟨hal-00851583⟩