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arXiv · 2005.10972

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

Zhiyuan Geng·Fanghua Lin
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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 $\Omega$.

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Zhiyuan Geng, Fanghua Lin. 2020-05-22. Large $m$ asymptotics for minimal partitions of the Dirichlet eigenvalue. https://doi.org/10.1007/s11425-020-1802-6

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