Friedrichs type inequalities in arbitrary domains

TL;DR

This paper establishes Friedrichs type inequalities for Sobolev functions in arbitrary domains, providing optimal constants independent of domain geometry, and extends results to vector-valued functions with symmetric gradients.

Contribution

It introduces new Friedrichs inequalities with optimal constants applicable to any domain, including vector-valued functions, based on general criteria from previous work.

Findings

Inequalities involve optimal norms and constants independent of domain shape.

Parallel inequalities are derived for vector-valued Sobolev spaces with symmetric gradients.

Results are based on general criteria from earlier research.

Abstract

First and second-order inequalities of Friedrichs type for Sobolev functions in arbitrary domains are offered. The relevant inequalities involve optimal norms and constants that are independent of the geometry of the domain. Parallel inequalities for symmetric gradient Sobolev spaces of vector-valued functions are also presented. The results are derived via general criteria established in our earlier contributions [4] and [5].