Friedrichs inequality in irregular domains

TL;DR

This paper extends Friedrichs and Gaffney inequalities to irregular domains in two and three dimensions by adapting Jones's methods to a generalized setting.

Contribution

It introduces a generalized version of Friedrichs and Gaffney inequalities applicable to irregular domains using adapted analytical techniques.

Findings

Proved Friedrichs inequality in irregular domains.

Extended Gaffney inequality to new domain classes.

Demonstrated the effectiveness of Jones's methods in this context.

Abstract

We prove a generalized version of Friedrichs and Gaffney inequalities for a bounded $(\varepsilon,\delta)$ domain $\Omega\subset\mathbb{R}^n$, $n=2,3$, by adapting the methods of Jones to our framework.