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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 \(k\geqslant 2\).

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