From bubbles to clusters: Multiple solutions to the Allen--Cahn system

TL;DR

This paper demonstrates the existence of multiple solutions to the multi-phase Allen-Cahn system on parallelizable Riemannian manifolds, linking solutions to topological invariants and isoperimetric profiles, especially as temperature approaches zero.

Contribution

It extends previous results to arbitrary numbers of phases using volume-fixing variations, providing new insights into phase separation on complex manifolds.

Findings

Number of solutions bounded by topological invariants

Solutions concentrate in isoperimetric-like regions as temperature decreases

Method applicable to any number of phases with small volume constraints

Abstract

We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed Riemannian manifolds by considering an arbitrary number of phases. Specifically, we show that on parallelizable manifolds, the number of solutions is bounded from below by topological invariants of the underlying manifold, provided the temperature parameter and volume constraint are sufficiently small. The Allen-Cahn system naturally arises in phase separation models, where solutions represent the distribution of distinct phases in a multi-component mixture. As the temperature parameter approaches zero, the system's energy approximates the multi-isoperimetric profile, leading to solutions concentrating in regions resembling isoperimetric clusters. For two or three phases, these results rely on classifying isoperimetric clusters. However, this classification remains incomplete for a larger number of phases. To address this technical issue, we employ a "volume-fixing variations" approach, enabling us to establish our results for any number of phases and small volume constraints. This offers deeper insights into phase separation phenomena on manifolds with arbitrary geometry.