Symmetry breaking in a constrained Cheeger type isoperimetric inequality

TL;DR

This paper investigates the optimal constants in a Sobolev inequality for BV functions, characterizing the shape of domains that minimize the embedding constant based on domain measure.

Contribution

It introduces a shape optimization framework for Sobolev inequalities involving BV functions and fully characterizes the optimal domains for the inequality.

Findings

Complete characterization of optimal domains

Identification of symmetry-breaking phenomena

Explicit formulas for optimal constants

Abstract

We study the optimal constant in a Sobolev inequality for BV functions with zero mean value and vanishing outside a bounded open set. We are interested in finding the best possible embedding constant in terms of the measure of the domain alone. We set up an optimal shape problem and we completely characterize the behavior of optimal domains.