Abstract
For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška–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 the Friedrichs 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, Zsuppán [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 Babuška–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.
Mathematics Subject Classification (2010). 30A10, 35Q35.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Costabel, M. (2017). Inequalities of Babuška–Aziz and Friedrichs–Velte for Differential Forms. In: Maz'ya, V., Natroshvili, D., Shargorodsky, E., Wendland, W. (eds) Recent Trends in Operator Theory and Partial Differential Equations. Operator Theory: Advances and Applications, vol 258. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-47079-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-47079-5_4
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-47077-1
Online ISBN: 978-3-319-47079-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)