Inequalities of Babu\v{s}ka-Aziz and Friedrichs-Velte for differential forms

TL;DR

This paper generalizes inequalities related to differential forms, establishing their equivalence and constant equality across various dimensions without regularity or topological assumptions, extending prior results in the field.

Contribution

It introduces a unified framework for Friedrichs and Babuška–Aziz inequalities for differential forms in arbitrary dimensions, removing regularity constraints.

Findings

Proves equivalence of inequalities without domain regularity assumptions

Establishes equality of associated constants in general settings

Generalizes Friedrichs inequality for conjugate harmonic forms across dimensions

Abstract

For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babu{\v s}ka --Aziz for right inverses of the divergence and of Friedrichs on conjugate harmonic functions was shown by Horgan and Payne in 1983 [7]. In a previous paper [4] we proved that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. In three dimensions, Velte [9] studied a generalization of the notion of conjugate harmonic functions and corresponding generalizations of Friedrich's inequality, and he showed for sufficiently smooth simply-connected domains the equivalence with inf-sup conditions for the divergence and for the curl. For this equivalence, Zsupp{\'a}n [10] observed that our proof can be adapted, proving the equality between the corresponding constants without regularity assumptions on the domain. Here we formulate a generalization of the Friedrichs inequality for conjugate harmonic differential forms on bounded open sets in any dimension that contains the situations studied by Horgan--Payne and Velte as special cases. We also formulate the corresponding inf-sup conditions or Babu{\v s}ka --Aziz inequalities and prove their equivalence with the Friedrichs inequalities, including equality between the corresponding constants. No a-priori conditions on the regularity of the open set nor on its topology are assumed.