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 $\theta \in (0,\pi /2)$, 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 $k\u2a7e2$.

@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{\textendash}Cahn} equation in the plane}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {761--781}, publisher = {Elsevier}, volume = {29}, number = {5}, year = {2012}, doi = {10.1016/j.anihpc.2012.04.003}, zbl = {1254.35219}, mrnumber = {2971030}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.003/} }

TY - JOUR AU - Kowalczyk, Michał AU - Liu, Yong AU - Pacard, Frank TI - The space of 4-ended solutions to the Allen–Cahn equation in the plane JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 761 EP - 781 VL - 29 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.003/ UR - https://zbmath.org/?q=an%3A1254.35219 UR - https://www.ams.org/mathscinet-getitem?mr=2971030 UR - https://doi.org/10.1016/j.anihpc.2012.04.003 DO - 10.1016/j.anihpc.2012.04.003 LA - en ID - AIHPC_2012__29_5_761_0 ER -

%0 Journal Article %A Kowalczyk, Michał %A Liu, Yong %A Pacard, Frank %T The space of 4-ended solutions to the Allen–Cahn equation in the plane %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 761-781 %V 29 %N 5 %I Elsevier %U https://doi.org/10.1016/j.anihpc.2012.04.003 %R 10.1016/j.anihpc.2012.04.003 %G en %F 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/articles/10.1016/j.anihpc.2012.04.003/

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