Minimal partitions for $p$-norms of eigenvalues

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.