Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds

TL;DR

This paper proves the existence of multiple solutions to a vectorial Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds, using topological and variational methods.

Contribution

It establishes a lower bound on the number of solutions based on topological invariants and demonstrates the generic nondegeneracy of solutions for certain metrics.

Findings

Multiple solutions exist depending on manifold topology.

The ACH energy converges to weighted multi-perimeter, enabling variational analysis.

Generic metrics yield nondegenerate solutions.

Abstract

We prove the existence of multiple solutions to the Allen--Cahn--Hilliard (ACH) vectorial equation (with two equations) involving a triple-well (triphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on some topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy $\Gamma$-converges to the weighted multi-perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable setup to apply the photography method. Along the way, the lack of a closed analytic expression for the multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to the ACH system are nondegenerate.