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

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