Minimizers of the prescribed curvature functional in a Jordan domain with no necks

Abstract

We provide a geometric characterization of the minimal and maximal minimizer of the prescribed curvature functional $P(E)-\kappa |E|$ among subsets of a Jordan domain $\Omega$ with no necks of radius ${\kappa }^{-1}$, for values of $\kappa$ greater than or equal to the Cheeger constant of $\Omega$. As an application, we describe all minimizers of the isoperimetric profile for volumes greater than the volume of the minimal Cheeger set, relative to a Jordan domain $\Omega$ which has no necks of radius $r$, for all $r$. Finally, we show that for such sets and volumes the isoperimetric profile is convex.

arXiv comment24 pages, 4 figures

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Cited by in corpus (6)

• The Cheeger problem in abstract measure spaces
• The Optimal Hölder Exponent in Massari's Regularity Theorem
• Optimization of the anisotropic Cheeger constant with respect to the anisotropy
• Geometric criteria for the existence of capillary surfaces in tubes
• Isoperimetric sets and $p$-Cheeger sets are in bijection
• The isoperimetric problem in $2$d domains without necks