On the higher Cheeger problem

Abstract

We develop the notion of higher Cheeger constants for a measurable set Ω ⊂ R N . By the kth Cheeger constant we mean the valuewhere the infimum is taken over all k-tuples of mutually disjoint subsets of Ω, 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 h k (Ω) and spectral minimal k-partitions of Ω 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.

Escobar constants of planar domains

“…More precisely, the aim of this manuscript is to investigate the relation between I k (M) , k ≥ 3 , and the geometry of a planar domain M. We are in particular interested in the behavior of I k (M) for k large and the configuration of optimal k-tuples. This may share some similarities with the study of optimal Cheeger clusters, see [1,2,12] and reference therein for more details. A quantity similar to the second Escobar constant I 2 (M) also appears in the study of longtime existence result for the curve shortening flow [8,10].…”

“…The main motivation for the study of the higher Cheeger constants stems from the fact that they are used for bounding eigenvalues of the Laplace operator. 1 This relationship has been intensively studied in the literature, see [4,14,15] and the references therein.…”

Estimates on the spectral interval of validity of the anti-maximum principle

“…Finally, the following lemma is a consequence of the definition of the second eigenvalue, see, e.g., [7,Remark 4].…”

On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian

“…In [20] the authors defined a sequence of variational eigenvalues and proved that they can be approximated by the corresponding eigenvalues of the p-Laplacian as p → 1. The second variational eigenvalue of the 1-Laplacian can be characterized geometrically, as a consequence of [20,Theorem 2.4] and [21,Theorem 5.5] (see also [7]). In particular, if Ω = B 2 is a disc, it holds λ 2 (1; B 2 ) = λ (1; B 2 ), and therefore λ 2 (1; B 2 ) = λ 3 (1; B 2 ) = λ (1; B 2 ) by reasoning as in Proposition 3.1.…”