A Sufficient Criterion to Determine Planar Self-Cheeger Sets

We prove a sufficient criterion to determine if a planar set

minimizes the prescribed curvature functional

\mathcal{F}_\kappa[E]:=P(E)-\kappa|E| F κ [ E ] : = P ( E ) − κ ∣ E ∣

amongst

E\subset \Omega E ⊂ Ω

. As a special case, we derive a sufficient criterion to determine if

\Omega Ω

is a self-Cheeger set, i.e. if it minimizes the ratio

P(E)/|E| P ( E ) / ∣ E ∣

among all of its subsets. As a side effect we provide a way to build self-Cheeger sets.