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

Abstract

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

This paper has been accepted for publication in SCIENCE CHINA Mathematics