Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms

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{á}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.