Phase field approach to optimal packing problems and related Cheeger clusters

TL;DR

This paper investigates the asymptotic behavior of optimal clusters related to Cheeger constants in fixed domains, showing convergence to packing problems, and introduces a phase field method for computing these clusters with numerical validation.

Contribution

It presents a novel phase field approach based on Gamma convergence to compute Cheeger constants and optimal clusters, linking them to packing problems as parameters vary.

Findings

Optimal Cheeger clusters converge to packing solutions as parameters approach critical values.

The phase field method efficiently computes Cheeger constants and clusters in 2D and 3D.

Numerical experiments validate the theoretical asymptotic and computational results.

Abstract

In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the "critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions.