On connections between domain specific constants in some norm inequalities

TL;DR

This paper explores relationships between key constants in various fundamental inequalities related to harmonic functions, divergence, and rotation, providing new insights into their interconnectedness in mathematical analysis.

Contribution

It establishes novel connections between domain-specific constants in Friedrichs-Velte, Babuška-Aziz, and Poincaré inequalities, including an improved inequality for rotation.

Findings

Connections between constants in different inequalities are derived.

An improved Poincaré inequality for rotation is established.

The method applies to spatial domains, enhancing existing inequalities.

Abstract

We derive connections between optimal domain specific constants figuring in the Friedrichs-Velte inequality for conjugate harmonic functions, in the Babu\v{s}ka-Aziz inequality for the divergence and in the improved Poincar\'e inequality for the gradient. With the same method we obtain for spatial domains an improved Poincar\'e inequality for the rotation in connection with the corresponding Babu\v{s}ka-Aziz inequality.