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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 \(j=0,\ldots,2k-1\).

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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