# On the higher Cheeger problem

## TL;DR

This paper introduces higher Cheeger constants for measurable sets, explores their properties, and relates them to spectral minimal partitions, with applications to specific planar domains.

## Contribution

It defines higher Cheeger constants, proves existence of minimizers with special properties, and connects these constants to spectral partition problems.

## Key findings

- Existence of minimizers for higher Cheeger constants.
- Relation between higher Cheeger constants and spectral minimal partitions.
- Application to determine the second Cheeger constant of planar domains.

## Abstract

We develop the notion of higher Cheeger constants for a measurable set $\Omega \subset \mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value \[h_k(\Omega) = \inf \max \{h_1(E_1), \dots, h_1(E_k)\},\] where the infimum is taken over all $k$-tuples of mutually disjoint subsets of $\Omega$, and $h_1(E_i)$ is the classical Cheeger constant of $E_i$. We prove the existence of minimizers satisfying additional "adjustment" conditions and study their properties. A relation between $h_k(\Omega)$ and spectral minimal $k$-partitions of $\Omega$ associated with the first eigenvalues of the $p$-Laplacian under homogeneous Dirichlet boundary conditions is stated. The results are applied to determine the second Cheeger constant of some planar domains.