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Multiple-end solutions to the Allen-Cahn equation in $$\mathbb R^2$$. (English) Zbl 1203.35108
The authors study the construction of a new class of solutions, in the entire plane $${\mathbb R}^2$$, for the semilinear elliptic equation $\Delta u + (1-u^2)u = 0,$ which is known as the Allen–Cahn equation. This equation originates from the gradient theory of phase transitions [S. Allen and J.W. Cahn, Acta Metall. 27, 1084–1095 (1979)], and it also has connections to the theory of minimal hypersurfaces [see F. Pacard and M. Ritoré, J. Differ. Geom. 64, No. 3, 359–423 (2003; Zbl 1070.58014)]. They show that for given $$k \geq 1$$, there exists a family of solutions whose zero level sets are, away from a compact set, asymptotic to $$2k$$ straight lines which are called ends. Furthermore, these solutions have the property that there exist $\theta_0 < \theta_1 < \cdots < \theta_{2k} = \theta_0 + 2\pi$ such that $\lim_{r\to + \infty}u(re^{i\theta}) = (-1)^j$ uniformly in $$\theta$$ on compact subsets of the interval $$(\theta_j,\theta_{j+1})$$ for $$j=0,\ldots,2k-1$$.

##### MSC:
 35J61 Semilinear elliptic equations 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35A30 Geometric theory, characteristics, transformations in context of PDEs
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##### References:
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