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

João Henrique de Andrade, Dario Corona, Stefano Nardulli, Paolo Piccione, Raoní Ponciano

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.

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

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.

Paper Structure

This paper contains 6 sections, 12 theorems, 142 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Tools and Preliminaries
  3. Outline for the Photography Method
  4. Photography Map
  5. Barycenter Map
  6. Proof of the Main Result

Key Result

Theorem A

Let $N$ and $m$ be two integers such that $N \ge 2$ and $m \ge 1$. Let $(M,g)$ be an $N$-dimensional closed and parallelizable Riemannian manifold let $W\in C^2(\mathbb{R}^m,\mathbb{R}_+)$ a multi-well potential. For every $\mathrm{v} \in \mathbb{R}^m_{>0}$ there exists $\alpha^* = \alpha^*(M,g,\mat

Figures (3)

  • Figure 1: A graphic representation of a multi-well potential (see Definition \ref{['def:multi-well-potential']}), where $m = 2$ and the points $\{\mathbf z_0,\mathbf z_1,\mathbf z_2\}$ form the vertices of an equilateral triangle.
  • Figure 2: The main idea for the construction of the photography map. In this figure, we have $N = 2$ and $m = 7$.
  • Figure 3: A graphic representation of the construction for the proof of Lemma \ref{['lem:fixing-volumes']}. In each interface, two little balls can be constructed to "inflate" or "deflate" the chamber. If the chamber is not linked with the exterior one, multiple steps are required to obtain the desired volume.

Theorems & Definitions (39)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Example 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem A
  • Remark 1.8
  • Theorem B: MR1322324MR1384393
  • ...and 29 more