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

Phase field approach to optimal packing problems and related Cheeger clusters