Generalized perimeters and gradient estimates

TL;DR

This paper introduces a unified variational framework for generalized perimeters, deriving isoperimetric inequalities and gradient estimates, and establishing new connections between Cheeger's inequality and transport inequalities.

Contribution

It develops an abstract framework that unifies existing results on perimeters, Cheeger's inequalities, and gradient estimates, and reveals new implications between these concepts.

Findings

Unified framework for generalized perimeters and inequalities

Recovery of the $W_1-W^{1,1}$ transport inequality

Cheeger's inequality implies Calderón-Zygmund gradient estimates

Abstract

We use a variational formulation to define a generalized notion of perimeter, from which we derive abstract isoperimetric Cheeger's inequalities via gradient estimates on solutions of Poisson equations. Our abstract framework unifies many existing results and in particular allows us to recover the $W_1-W^{1,1}$ transport inequality, which strengthens the usual transport-information inequality. Conversely, we also prove that Cheeger's inequality implies certain first order Calder{\'o}n-Zygmund-type gradient estimates.