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The space of 4-ended solutions to the Allen-Cahn equation in the plane. (English) Zbl 1254.35219
Summary: We are interested in entire solutions of the Allen-Cahn equation $$\Delta u - F^{\prime}(u)=$$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 $$\theta , \pi - \theta , \pi +\theta$$ 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 $$k\geqslant 2$$.

##### MSC:
 35Q82 PDEs in connection with statistical mechanics 35J61 Semilinear elliptic equations 35B08 Entire solutions to PDEs
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