Fuglede-Type Arguments for Isoperimetric Problems and Applications to Stability Among Convex Shapes

.This paper is concerned with the stability of the ball for a class of isoperimetric problems under a convexity constraint. Considering the problem of minimizing \(P+ε R\) among convex subsets of \(R^N\) of fixed volume, where \(P\) is the perimeter functional, \(R\) is a perturbative term, and \(ε > 0\) is a small parameter, the stability of the ball for this perturbed isoperimetric problem means that the ball is the unique (local, up to translation) minimizer for any \(ε\) sufficiently small. We investigate independently two specific cases where \(Ω ↦ R(Ω )\) is an energy arising from PDE theory, namely, the capacity and the first Dirichlet eigenvalue of a domain \(Ω ⊂ R^N\). While in both cases stability fails among all shapes, in the first case we prove the (nonsharp) stability of the ball among convex shapes, by building an appropriate competitor for the capacity of a perturbation of the ball. In the second case we prove the sharp stability of the ball among convex shapes by providing the optimal range of \(ε\) such that stability holds, relying on the selection principle technique and a regularity theory under convexity constraint.Keywordsshape optimizationisoperimetric problemstabilityMSC codes49Q1049K2046E3535P15