This paper stems from the idea of adopting a new approach to solve some classical optimal packing problems for balls. In fact, we attack this kind of problems (which are of discrete nature) by means of shape optimization techniques, applied to suitable \(\Gamma \)-converging sequences of energies associated to Cheeger type problems. More precisely, in a first step we prove that different optimal packing problems are limits of sequences of optimal clusters associated to the minimization of energies involving suitable (generalized) Cheeger constants. In a second step, we propose an efficient phase field approach based on a multiphase \(\Gamma \)-convergence result of Modica–Mortola type, in order to compute those generalized Cheeger constants, their optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions. Our continuous shape optimization approach to solve discrete packing problems circumvents the NP-hard character of these ones, and efficiently leads to configurations close to the global minima.

中文翻译：

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最优装箱问题和相关Cheeger簇的相场方法

本文源于采用一种新方法来解决一些经典的球最佳包装问题的想法。实际上，我们通过形状优化技术来攻击这类问题（具有离散性质），并将其应用于适合的与Cheeger型问题相关的能量的（Gamma）收敛序列。更准确地说，在第一步中，我们证明了不同的最佳堆积问题是与涉及合适（广义）Cheeger常数的能量最小化相关的最佳簇的序列的极限。在第二步中，我们提出了一种基于多相\（\ Gamma \）的有效相场方法-Modica–Mortola类型的收敛结果，以便计算那些广义的Cheeger常数，它们的最佳聚类以及作为渐近结果的结果是最佳装箱。在两个和三个空间维度上进行了数值实验。我们的连续形状优化方法可解决离散包装问题，从而规避了这些问题的NP-hard特性，并有效地使配置接近全局最小值。