Abstract
To measure data and solutions spatially, we recall a number of useful definitions and results on Lebesgue and standard Sobolev spaces. Then, we introduce more specialized Sobolev spaces, which are better suited to measuring solutions to electromagnetics problems, in particular, the divergence and the curl of fields. This also allows one to measure their trace at interfaces between two media, or on the boundary. Last, we construct ad hoc function spaces, adapted to the study of time- and space-dependent electromagnetic fields.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Given any subset S of \(\mathbb {R}^n\), 1 S denotes the indicator function of S .
- 2.
The space \({\mathcal {D}}(\varOmega )\) can also be denoted by \(C^\infty _c(\varOmega )\), where the index c stands for compact support.
- 3.
Classically, for \(k\in \mathbb {N}\), β ∈ ]0, 1], \({\mathcal O}\subset \mathbb {R}^n\), \(C^{k,\beta }({\mathcal O})\) is the Hölder space defined by
$$\displaystyle \begin{aligned} C^{k,\beta}({\mathcal O}):=\{f\in C^{k}({\mathcal O})\ :\ \sum_{\alpha\in\mathbb{N}^n,\ |\alpha|=k}\sup_{{\boldsymbol{x}}\ne\boldsymbol{y}}\frac{|\partial_\alpha f({\boldsymbol{x}})-\partial_\alpha f(\boldsymbol{y})|}{|{\boldsymbol{x}}-\boldsymbol{y}|{}^\beta}<\infty\},\end{aligned}$$where \(C^k({\mathcal O}):=\{f\in C^0({\mathcal O})\ :\ \partial _\alpha f\in C^0({\mathcal O}),\ \forall \alpha \in \mathbb {N}^n,\ |\alpha |\le k\}\).
Lipschitz-continuity coincides with C 0, 1 continuity.
- 4.
Given any subset S of \(\mathbb {R}^n\), int(S) denotes the interior of S.
- 5.
Evidently, a direct construction is also possible!
- 6.
References
R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975)
R.A. Adams, J.J.F. Fournier, Sobolev Spaces, 2nd edn. (Academic Press, New York, 2003)
C. Amrouche, C. Bernardi, M. Dauge, V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)
C. Bernardi, M. Dauge, Y. Maday, Compatibility of traces on the edges and corners of a polyhedron. C. R. Acad. Sci. Paris Ser. I 331, 679–684 (2000)
J. Bourgain, H. Brezis, P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, ed. by J.L. Menaldi et al. (IOP Press, 2001), pp. 439–455
H. Brezis, Analyse fonctionnelle. Théorie et applications (Masson, Paris, 1983). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext (Springer, 2011)
A. Buffa, P. Ciarlet, Jr., On traces for functional spaces related to Maxwell’s equations. Part I: an integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24, 9–30 (2001)
M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, vol. 1341 (Springer, Berlin, 1988)
R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology (Springer, Berlin, 1990)
P. Fernandes, G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods App. Sci. 7, 957–991 (1997)
E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. di Padova 27, 284–305 (1957)
J. Garsoux, Espaces Vectoriels Topologiques et Distributions (Dunod, Paris, 1963)
V. Girault, P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, vol. 5 (Springer, Berlin, 1986)
P. Grisvard, Théorèmes de traces relatifs à un polyèdre. C. R. Acad. Sci. Paris Ser. I 278, 1581–1583 (1974)
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24 (Pitman, London, 1985)
P. Grisvard, Singularities in Boundary Value Problems. RMA, vol. 22 (Masson, Paris, 1992)
J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1 (Dunod, Paris, 1968)
J. Nečas, Les méthodes directes en théorie des équations elliptiques (Masson, Paris, 1967)
P.-A. Raviart, J.-M. Thomas, Introduction à l’analyse numérique des équations aux dérivées partielles (Masson, Paris, 1983)
L. Schwartz, Théorie des Distributions (Hermann, Paris, 1966)
L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3 (Springer, 2007)
F. Trèves, Basic Linear Partial Differential Equations (Academic Press, New York, 1975)
K. Yosida, Functional Analysis (Springer, Berlin, 1980)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Assous, F., Ciarlet, P., Labrunie, S. (2018). Basic Applied Functional Analysis. In: Mathematical Foundations of Computational Electromagnetism. Applied Mathematical Sciences, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-70842-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-70842-3_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-70841-6
Online ISBN: 978-3-319-70842-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)