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

Kowalczyk, Michał ; Liu, Yong ; Pacard, Frank

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 761-781 / Harvested from Numdam

Résumé

We are interested in entire solutions of the Allen–Cahn equation $ \mathrm{\Delta }u-{F}^{\prime }(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 \in (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 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$.

Détail

Publié le : 2012-01-01

MR 2971030 | Zbl 1254.35219

DOI : https://doi.org/10.1016/j.anihpc.2012.04.003

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