Saddle solutions to Allen-Cahn equations in doubly periodic media
Francesca Alessio, Changfeng Gui, Piero Montecchiari · Indiana University Mathematics Journal · 2016
We consider a class of periodic Allen-Cahn equations $ -Delta u(x,y)+a(x,y)W'(u(x,y))=0,quad (x,y)inR^{2}$ where $ain C(R^2)$ is an even, periodic, positive function represenitng a doubly periodic media and $W:R oR$ 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 axises 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 media.