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

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 volume constraint is sufficiently small. This system naturally arises in phase separation models, where solutions represent the distribution of distinct phases in a multi-component mixture. As the regularization parameter approaches zero, the system's energy approximates the multi-isoperimetric profile, and solutions concentrate in regions resembling isoperimetric clusters. For two or three phases, these results rely on classifying isoperimetric clusters, which is incomplete for a larger number of phases. To address this issue, we employ the “volume-fixing variations” approach, enabling us to establish results for any number of phases. This offers more profound insights into phase separation phenomena on manifolds with arbitrary geometry.