On the higher Cheeger problem

Vladimir Bobkov (1) , Enea Parini (2)

1 University of West Bohemia - Department of Mathematics
2 I2M - Institut de Mathématiques de Marseille

We develop the notion of higher Cheeger constants for a measurable set RN. By the kth Cheeger constant we mean the value where the infimum is taken over all k-tuples of mutually disjoint subsets of , and h1(Ei) is the classical Cheeger constant of Ei. We prove the existence of minimizers satisfying additional adjustment' conditions and study their properties. A relation between hk() and spectral minimal k-partitions of 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.

Mots clés: SPECTRAL MINIMAL PARTITIONS; P-LAPLACIAN; EIGENVALUES; SETS; APPROXIMATION; PERIMETER; OPERATOR

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Vladimir Bobkov, Enea Parini. On the higher Cheeger problem. Journal of the London Mathematical Society, 2018, 97 (3), pp.575-600. ⟨10.1112/jlms.12119⟩. ⟨hal-02060838⟩