A unified divergent approach to Hardy-Poincar\'e inequalities in classical and variable Sobolev spaces

TL;DR

This paper introduces a unified method to derive Hardy-Poincaré inequalities applicable to classical and variable Sobolev spaces, providing a general framework that simplifies proofs and offers insights into optimal constants.

Contribution

The authors develop a novel, unified approach to Hardy-Poincaré inequalities that encompasses classical and variable exponent Sobolev spaces without relying on compactness assumptions.

Findings

Derives a general Hardy-Poincaré inequality from which classical inequalities follow

Applicable to both bounded and unbounded domains

Provides geometric insights into the best constants

Abstract

We present a unified strategy to derive Hardy-Poincar\'e inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincar\'e inequality from which the classical Poincar\'e and Hardy inequalities immediately follow. The idea also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.