IUMJ

Title: Saddle solutions to Allen-Cahn equations in doubly periodic media

Authors: Francesca Alessio, Changfeng Gui and Piero Montecchiari

Issue: Volume 65 (2016), Issue 1, 199-221

Abstract:

We consider a class of periodic Allen-Cahn equations \begin{equation}\tag{$1$} -\Delta u(x,y)+a(x,y)W'(u(x,y))=0,\quad (x,y)\in\mathbb{R}^2, \end{equation} where $a\in C(\mathbb{R}^2)$ is an even, periodic, positive function representing a doubly periodic media, and $W:\mathbb{R}\to\mathbb{R}$ is a classical double well potential such as the Ginzburg-Landau potential $W(s)=(s^2-1^2)^2$. We show the existence and asymptotic behavior of a saddle solution on the entire plane, which has odd symmetry with respect to both axes, and even symmetry with respect to the line $x=y$. This result generalizes the classic result on saddle solutions of Allen-Cahn equation in a homogeneous medium.