Regularity in shape optimization under convexity constraint

TL;DR

This paper establishes C1,1 regularity of shape optimizers under convexity constraints for a class of isoperimetric problems, using a novel quasi-minimizer framework and a cutting procedure.

Contribution

It introduces a new notion of quasi-minimizer for convex shapes and proves their regularity, extending regularity results to volume-constrained problems and PDE-related perturbations.

Findings

Minimizers are proven to be C1,1-regular.

The regularity result applies to PDE-type perturbations.

A counterexample shows higher regularity cannot be generally expected.

Abstract

This paper is concerned with the regularity of shape optimizers of a class of isoperimetric problems under convexity constraint. We prove that minimizers of the sum of the perimeter and a perturbative term, among convex shapes, are C 1,1-regular. To that end, we define a notion of quasi-minimizer fitted to the convexity context and show that any such quasi-minimizer is C 1,1-regular. The proof relies on a cutting procedure which was introduced to prove similar regularity results in the calculus of variations context. Using a penalization method we are able to treat a volume constraint, showing the same regularity in this case. We go through some examples taken from PDE theory, that is when the perturbative term is of PDE type, and prove that a large class of such examples fit into our C 1,1-regularity result. Finally we provide a counterexample showing that we cannot expect higher regularity in general.