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 + epsilon R among convex subsets of R-N of fixed volume, where P is the perimeter functional, R is a perturbative term, and epsilon > 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 epsilon sufficiently small. We investigate independently two specific cases where Omega (sic) R(Omega) is an energy arising from PDE theory, namely, the capacity and the first Dirichlet eigenvalue of a domain Omega subset of 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 epsilon such that stability holds, relying on the selection principle technique and a regularity theory under convexity constraint.