Submitted Paper

Inserted: 20 jun 2017
Last Updated: 20 jun 2017

Year: 2017

Abstract:

In a fixed domain of $\R ^N$ 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.

Keywords: Cheeger constant, optimal packing, phase field, Modica-Mortola

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• Packing.pdf