Abstract
We now begin the theoretical study of elliptic boundary value problems in a context that is more general than the one-dimensional model problem treated in Chap. 1. We will focus on one approach, which is called the variational approach. There are other ways of solving elliptic problems, such as working with Green functions as seen in Chap. 2. The variational approach is quite simple and well suited for a whole class of approximation methods, as we will see later.
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- 1.
The Laplacian is a specific linear combination of some of the second order derivatives. So it being in \(L^2\) is a priori less than all individual second order derivatives, even those not appearing in the Laplacian, being in \(L^2\), except when \(d=1\).
- 2.
The space \(H^{3/2}(\partial \varOmega )\) is the space of traces of \(H^2(\varOmega )\) functions.
- 3.
This condition is also sometimes called coerciveness.
- 4.
Since we already know it is one-to-one, it will then be an isomorphism.
- 5.
This is a pretty common strategy, to be kept in mind.
- 6.
If we had worked with the semi-norm for the Dirichlet problem, we would have had to do the continuity all over again here...
- 7.
We have always said the variational formulation, but there is no evidence that it is unique in general.
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© 2016 Springer International Publishing Switzerland
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Le Dret, H., Lucquin, B. (2016). The Variational Formulation of Elliptic PDEs. 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_4
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DOI: https://doi.org/10.1007/978-3-319-27067-8_4
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Publisher Name: Birkhäuser, Cham
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