Calculus of Variations and Geometric Measure Theory
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G. P. Leonardi - G. Saracco

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

created by saracco on 20 Dec 2019
modified on 18 May 2020

[BibTeX]

Published Paper

Inserted: 20 dec 2019
Last Updated: 18 may 2020

Journal: ESAIM Control Optim. Calc. Var.
Year: 2020
Doi: 10.1051/cocv/2020030

ArXiv: 1912.09462 PDF

Abstract:

We provide a geometric characterization of the minimal and maximal minimizer of the prescribed curvature functional $P(E)-\kappa\, \textrm{vol}(E)$ 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.

Keywords: Cheeger constant, prescribed mean curvature, perimeter minimizer


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