Abstract
In order to go beyond the somewhat naive existence theory and finite difference method of approximation of elliptic boundary value problems seen in Chaps. 1 and 2, we need to develop a more sophisticated point of view. This requires in turn some elements of analysis pertaining to function spaces in several variables, starting with some abstract Hilbert space theory. This is the main object of this chapter. As already mentioned in the preface, this chapter can be read quickly at first, for readers who are not too interested in the mathematical details and constructions therein. A summary of the important results needed for the subsequent chapters is thus provided at the end of the chapter.
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Notes
- 1.
This is doubly false if \(\varOmega \) is not bounded.
- 2.
This particular assumption is only because this is the context in which we will use partitions of unity here. It should be clear from the proof, that the result extends to more general covers.
- 3.
Note that there is no regularity or boundedness hypothesis made on \(\varOmega \) itself.
- 4.
It is enough for Poincaré’s inequality to be valid that \(\varOmega \) be included in such a strip although not necessarily bounded.
- 5.
And not only almost everywhere, since we are now talking about the continuous representative of v.
- 6.
The fairly easy proof uses the density of continuous, compactly supported functions in \(L^2({\mathbb R}^d)\).
- 7.
Instead of upwards for the density result.
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© 2016 Springer International Publishing Switzerland
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Le Dret, H., Lucquin, B. (2016). A Review of Analysis. In: Partial Differential Equations: Modeling, Analysis and Numerical Approximation. International Series of Numerical Mathematics, vol 168. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27067-8_3
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DOI: https://doi.org/10.1007/978-3-319-27067-8_3
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