Minimal Partitions for p-norms of Eigenvalues

TL;DR

This paper investigates optimal partitions of geometric shapes to minimize p-norms of Dirichlet-Laplace eigenvalues, introducing three numerical algorithms that outperform existing bounds and analyzing how minimal partitions vary with p.

Contribution

The paper introduces three numerical algorithms for approximating minimal partitions that optimize p-norms of eigenvalues, providing tighter bounds and insights into their behavior across different p-values.

Findings

Algorithms outperform theoretical bounds

Minimal partitions differ for various p-values

Numerical results reveal partition behaviors

Abstract

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.