Saddle solutions to Allen-Cahn equations in doubly periodic media Francesca AlessioChangfeng GuiPiero Montecchiari 35J6035B0535B4035J2034C37Heteroclinic solutionsSaddle SolutionsElliptic EquationsVariational Methods 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. Indiana University Mathematics Journal 2016 text pdf 10.1512/iumj.2016.65.5772 10.1512/iumj.2016.65.5772 en Indiana Univ. Math. J. 65 (2016) 199 - 221 state-of-the-art mathematics http://iumj.org/access/