The Cheeger problem in abstract measure spaces. (English) Zbl 1536.49036

Franceschi, Valentina; Pinamonti, Andrea; Saracco, Giorgio; Stefani, Giorgio

J. Lond. Math. Soc., II. Ser. 109, No. 1, Article ID e12840, 55 p. (2024).

The Cheeger constant of a set \(\Omega\subset \mathbb R^n\) is given by \[ h(\Omega)=\inf\left\{\frac{P(E)}{|E|} : E\subset \Omega,\,|E|>0\right\} \] where \(P(E)\) is a variational perimeter of \(E\) and \(|E|\) is the \(n\)-dimensional Lebesgue measure of \(E\). A lot of results are known as well as numerous applications, even for the problem on general measure spaces. The authors want to study the Cheeger problem on general measure spaces, instead of \(\mathbb R^n\), and a correspondingly more general notion of perimeter. In this paper the authors are interested in finding minimal assumptions on the space and on the perimeter functional in such a way it is possible to prove some of the most important properties of Cheeger sets: existence, link to sets with prescribed mean curvature, link to the first eigenvalue of the Dirichlet 1-Laplacian, link to the Dirichlet \(p\)-Laplacian and \(p\)-torsion.

Reviewer: Luca Lussardi (Torino)

MSC:

49Q20	Variational problems in a geometric measure-theoretic setting
35P15	Estimates of eigenvalues in context of PDEs
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

Cheeger sets; eigenvalues; p-Laplacian

References:

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