Multiple-end solutions to the Allen-Cahn equation in \(\mathbb R^2\) (Q1048172)

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 [\textit{S. Allen} and \textit{J.W. Cahn}, Acta Metall. 27, 1084--1095 (1979)], and it also has connections to the theory of minimal hypersurfaces [see \textit{F. Pacard} and \textit{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 \(j=0,\ldots,2k-1\).