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On the higher Cheeger problem. (English) Zbl 1394.49041
Summary: 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 radially symmetric planar domains.

MSC:
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q10 Optimization of shapes other than minimal surfaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q20 Variational problems in a geometric measure-theoretic setting
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