Analysis course, ENPC first term
Allmost all of the scientific and engineering fields that are taught
at the ENPC include models based on partial differential equations
(PDEs).
The aim of the course is to present some mathematical tools to deal with
this kind of equations. The aim is not to give a comprehensive overwiew
of the main technics of the field, but rather to completely study an
important case, the Poisson problem in a bounded open subset, with
Dirichlet boundary conditions. We study the following questions:
- Is there a solution?
- If so, is it unique?
The problem we study is
- a standard problem in PDEs analysis,
- a generic problem in engineering sciences,
- the problem chosen as an example in the Scientific Computing
course, in order to present the Finite Element Method.
Course outline:
- Banach spaces (Cauchy sequences, Picard fixed point theorem,
Cauchy-Lipschitz theorem),
- Hilbert spaces,
- Lebesgue integral and Lp spaces,
- Distributions (first exemples, Sobolev spaces H1
and H10, Poincaré lemma),
- Riesz theorem, Lax-Milgram theorem, variational formulation,
solving the Poisson problem,
- Fourier transformation.
Teaching team:
Eric Cancès, Alexandre
Ern, Jean-Frédéric Gerbeau, Régis Monneau and myself.
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Last update: March 2006.