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

TL;DR

This paper characterizes the extremal minimizers of a prescribed curvature functional within Jordan domains without necks, revealing their geometric properties and implications for the isoperimetric profile.

Contribution

It provides a geometric characterization of extremal minimizers of the prescribed curvature functional in Jordan domains with no necks, extending understanding of shape optimization.

Findings

Identifies minimal and maximal minimizers of the curvature functional

Describes all minimizers of the isoperimetric profile for large volumes

Shows convexity of the isoperimetric profile in these domains

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.