Rigidity results for a triple junction solution of Allen-Cahn system

TL;DR

This paper proves that in a 2D Allen-Cahn system with a triple-well potential, the solution's interfaces are asymptotically flat and well-approximated by 1D heteroclinic connections, extending previous symmetric results to the non-symmetric case.

Contribution

It generalizes existing symmetric triple junction results to the non-symmetric case by establishing asymptotic flatness and interface approximation properties.

Findings

Interfaces are asymptotically invariant and flat at infinity.

Diffuse interface remains close to the sharp interface within an O(1) neighborhood.

Results extend to non-symmetric triple junction solutions.

Abstract

For the two dimensional Allen-Cahn system with a triple-well potential, previous results established the existence of a minimizing solution $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ with a triple junction structure at infinity. We show that along each of three sharp interfaces, $u$ is asymptotically invariant in the direction of the interface and can be well-approximated by the 1D heteroclinic connections between two phases. Consequently, the diffuse interface is located in an $O(1)$ neighborhood of the sharp interface, and becomes nearly flat at infinity. This generalizes all the results for the triple junction solution with symmetry hypotheses to the non-symmetric case. The proof relies on refined sharp energy lower and upper bounds, alongside a precise estimate of the diffuse interface location.