Large $m$ asymptotics for minimal partitions of the Dirichlet eigenvalue

TL;DR

This paper investigates the asymptotic behavior of minimal partitions of a domain based on Dirichlet eigenvalues as the number of partitions grows large, establishing a shape-independent limit for the normalized sum.

Contribution

It proves the existence of a universal limit for the normalized sum of eigenvalues in large partitions, independent of the domain's shape.

Findings

The limit of the minimal normalized sum exists as m approaches infinity.

This limit is independent of the domain's shape.

The limit value is a constant c_0.

Abstract

In this paper, we study large $m$ asymptotics of the $l^1$ minimal $m$-partition problem for Dirichlet eigenvalue. For any smooth domain $\Omega\in \mathbb{R}^n$ such that $|\Omega|=1$, we prove that the limit $\lim\limits_{m\rightarrow\infty}l_m^1(\Omega)=c_0$ exists, and the constant $c_0$ is independent of the shape of $\Omega$. Here $l_m^1(\Omega)$ denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any $m$-partition of $\Omega$.