On the higher Cheeger problem

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.