Isoperimetric functional inequalities via the maximum principle: the exterior differential systems approach

TL;DR

This paper introduces a unified approach using exterior differential systems to prove classical isoperimetric inequalities and derives new sharpened inequalities, notably improving Beckner--Sobolev bounds with Gaussian measures.

Contribution

It establishes the first connection between isoperimetric inequalities and exterior differential systems, providing a new method to prove and generate inequalities.

Findings

Unified proof of classical inequalities like log-Sobolev and Beckner's.

New sharpened Beckner--Sobolev inequalities with Gaussian measure.

Reduction of isoperimetric problems to exterior differential systems.

Abstract

The goal of this note is to give the unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems studied by R.~Bryant and P.~Griffiths. In this note we list several classical by now isopereimetric inequalities which can be proved in a unified way. This unified approach reduces them to the so-called exterior differential systems studied by Robert Bryant and Phillip Griffiths. To the best of our knowledge, this is the first article where this connection is used. After reviewing a list of classical inequalities (log-Sobolev inequality, Beckner's inequality, Bobkov's functional isoperimetric inequality and several other inequalities) we use our method to generate new isoperimetric inequalities, in particular, we found the sharpening of Beckner--Sobolev inequalities with Gaussian measure. Key words: log-Sobolev inequality, Poincar\'e inequality, Bobkov's inequality, Gaussian isoperimetry, semigroups, maximum principle, Monge--Amp\`ere equation with drift, exterior differential systems, backwards heat equation, (B) theorem