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}, mrnumber = {2971030}, zbl = {1254.35219}, 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/ 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 http://www.numdam.org/articles/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/

[1] Entire solutions of semilinear elliptic equations in ${R}^{3}$ and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 no. 4 (2000), 725-739 | MR | Zbl

, ,[2] Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math. 50 no. 11 (1997), 1089-1111 | MR | Zbl

, , ,[3] X. Cabré, Uniqueness and stability of saddle-shaped solutions to the Allen–Cahn equation, arXiv e-prints, 2011. | MR

[4] Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys. 43 no. 6 (1992), 984-998 | MR | Zbl

, , ,[5] M. del Pino, M. Kowalczyk, F. Pacard, Moduli space theory for the Allen–Cahn equation in the plane, Trans. Amer. Math. Soc. (2010), in press. | MR

[6] Multiple-end solutions to the Allen–Cahn equation in ${R}^{2}$, J. Funct. Anal. 258 no. 2 (2010), 458-503 | MR | Zbl

, , , ,[7] On De Giorgiʼs conjecture in dimension $N\u2a7e9$, Ann. of Math. (2) 174 no. 3 (2011), 1485-1569 | MR | Zbl

, , ,[8] On the finiteness of the Morse index for Schrödinger operators, arXiv:1011.3390v2 (2011) | MR

,[9] On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 no. 1 (1985), 121-132 | EuDML | MR | Zbl

,[10] On a conjecture of De Giorgi and some related problems, Math. Ann. 311 no. 3 (1998), 481-491 | MR | Zbl

, ,[11] Hamiltonian identities for elliptic partial differential equations, J. Funct. Anal. 254 no. 4 (2008), 904-933 | MR | Zbl

,[12] C. Gui, Even symmetry of some entire solutions to the Allen–Cahn equation in two dimensions, arXiv e-prints, 2011. | MR

[13] Embedded minimal surfaces derived from Scherkʼs examples, Manuscripta Math. 62 no. 1 (1988), 83-114 | EuDML | MR | Zbl

,[14] Nondegeneracy of the saddle solution of the Allen–Cahn equation on the plane, Proc. Amer. Math. Soc. 139 no. 12 (2011), 4319-4329 | MR | Zbl

, ,[15] M. Kowalczyk, Y. Liu, F. Pacard, The classification of four ended solutions to the Allen–Cahn equation on the plane, in preparation, 2011.

[16] The classification of singly periodic minimal surfaces with genus zero and Scherk type ends, Trans. Amer. Math. Soc. 359 (2007), 965-990 | MR | Zbl

, ,[17] Regularity of at level sets in phase transitions, Ann. of Math. (2) 169 no. 1 (2009), 41-78 | MR | Zbl

,[18] On the stability of the saddle solution of Allen–Cahnʼs equation, Proc. Roy. Soc. Edinburgh Sect. A 125 no. 6 (1995), 1241-1275 | MR | Zbl

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