Multiplicity of layered solutions for Allen-Cahn systems with symmetric double well potential

TL;DR

This paper proves the existence of infinitely many layered solutions for a symmetric Allen-Cahn system in three dimensions, exhibiting dihedral symmetry and specific asymptotic behavior, using variational methods.

Contribution

It establishes the existence of infinitely many solutions with dihedral symmetry for the Allen-Cahn system with symmetric double well potential, under generic conditions.

Findings

Existence of infinitely many solutions with dihedral symmetry.

Solutions converge to minima uniformly in certain directions.

Characterization of solutions' asymptotic behavior at infinity.

Abstract

We study the existence of solutions $u:\R^{3}\to\R^{2}$ for the semilinear elliptic systems \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+\nabla W(u(x,y,z))=0, \end{equation} where $W:\R^{2}\to\R$ is a double well symmetric potential. We use variational methods to show, under generic non degenerate properties of the set of one dimensional heteroclinic connections between the two minima $\a_{\pm}$ of $W$, that (\ref{eq:abs}) has infinitely many geometrically distinct solutions $u\in C^{2}(\R^{3},\R^{2})$ which satisfy $u(x,y,z)\to \a_{\pm}$ as ${x\to\pm\infty}$ uniformly with respect to $(y,z)\in\R^{2}$ and which exhibit dihedral symmetries with respect to the variables $y$ and $z$. We also characterize the asymptotic behaviour of these solutions as $|(y,z)|\to +\infty$.