In this article we are interested in studying partitions of the square, the disk and the equilateral triangle which minimize a p-norm of eigenvalues of the Dirichlet-Laplace operator. The extremal case of the infinity norm, where we minimize the largest fundamental eigenvalue of each cell, is one of our main interests. We propose three numerical algorithms which approximate the optimal configurations and we obtain tight upper bounds for the energy, which are better than the ones given by theoretical results. A thorough comparison of the results obtained by the three methods is given. We also investigate the behavior of the minimal partitions with respect to p. This allows us to see when partitions minimizing the 1-norm and the infinity-norm are different.

中文翻译：

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$p$-特征值范数的最小分区

在本文中，我们感兴趣的是研究正方形、圆盘和等边三角形的分区，这些分区最小化 Dirichlet-Laplace 算子的特征值的 p 范数。无穷范数的极值情况，我们最小化每个单元格的最大基本特征值，是我们的主要兴趣之一。我们提出了三种近似最优配置的数值算法，我们获得了能量的严格上限，这比理论结果给出的要好。给出了通过三种方法获得的结果的彻底比较。我们还研究了关于 p 的最小分区的行为。这使我们能够看到最小化 1 范数和无穷范数的分区何时不同。