A new multicomponent Poincar\'e-Beckner inequality

TL;DR

This paper introduces a novel multicomponent Poincaré-Beckner inequality, linking entropy and entropy production for gradient flows in measure spaces, with a complex proof involving geometric measure theory.

Contribution

It presents a new vectorial inequality of Poincaré-Beckner type, expanding the mathematical understanding of entropy-related inequalities in measure spaces.

Findings

Proved a new multicomponent Poincaré-Beckner inequality.

Connected the inequality to entropy-entropy production in gradient flows.

Utilized advanced geometric measure theory techniques in the proof.

Abstract

We prove a new vectorial functional inequality of Poincar\'{e}-Beckner type. The inequality may be interpreted as an entropy-entropy production one for a gradient flow in the metric space of Radon measures. The proof uses subtle analysis of combinations of related super- and sub-level sets employing the coarea formula and the relative isoperimetric inequality.