On the higher Cheeger problem

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 \[h_k(Ω) = \inf \max \{h_1(E_1), \dots, h_1(E_k)\},\] 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.

References in corpus (2)

• The Cheeger constant of a Jordan domain without necks
• The Cheeger N-problem in terms of BV functions

Cited by in corpus (8)

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• Estimates on the spectral interval of validity of the anti-maximum principle
• The Pseudo-orthogonality for Graph $1$-Laplacian Eigenvectors and Applications to Higher Cheeger Constants and Data Clustering
• Isoperimetric sets and $p$-Cheeger sets are in bijection