Connections between optimal constants in some norm inequalities for differential forms

TL;DR

This paper improves Poincaré inequalities related to differential forms by estimating domain-specific constants and extends known estimates to higher dimensions for star-shaped domains.

Contribution

It introduces an improved Poincaré inequality linked with Babuška-Aziz and Friedrichs-Velte inequalities and generalizes Horgan-Payne estimates to higher dimensions.

Findings

Derived improved Poincaré inequality for differential forms.

Estimated optimal constants for specific inequalities based on domain properties.

Extended Horgan-Payne estimates to higher-dimensional star-shaped domains.

Abstract

We derive an improved Poincar\'e inequality in connection with the Babu\v{s}ka-Aziz and Friedrichs-Velte inequalities for differential forms by estimating the domain specific optimal constants figuring in the respective inequalities with each other provided the domain supports the Hardy inequality. We also derive upper estimates for the constants of a star-shaped domain by generalizing the known Horgan-Payne type estimates for planar and spatial domains to higher dimensional ones.