On the inequalities of Babu\v{s}ka-Aziz, Friedrichs and Horgan-Payne

TL;DR

This paper proves the equivalence of Babuška-Aziz and Friedrichs inequalities without domain regularity assumptions and investigates the validity of the Horgan-Payne inequality across different star-shaped domains.

Contribution

It establishes the equivalence of these inequalities without regularity constraints and analyzes the applicability of the Horgan-Payne inequality to various domains.

Findings

Equivalence of Babuška-Aziz and Friedrichs inequalities holds without regularity conditions.

Horgan-Payne inequality is valid for some, but not all, star-shaped domains.

A weaker universal inequality is proven to hold for all bounded star-shaped domains.

Abstract

The equivalence between the inequalities of Babu\v{s}ka-Aziz and Friedrichs for sufficiently smooth bounded domains in the plane has been shown by Horgan and Payne 30 years ago. We prove that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. For the Horgan-Payne inequality, which is an upper bound of the Friedrichs constant for plane star-shaped domains in terms of a geometric quantity known as the Horgan-Payne angle, we show that it is true for some classes of domains, but not for all bounded star-shaped domains. We prove a weaker inequality that is true in all cases.