Many models in physics, biology or engineering are based on Partial
Differential
Equations (PDEs). Mathematical analysis is crucial to understand the properties
of the solutions and thus to
design more accurate models and efficient simulation methods.
In this
series of lectures we shall present several tools of Functional
Analysis that are of frequent
use in all these aspects of modern Applied mathematics.
We address the questions of the existence, uniqueness, and qualitative
properties of the solutions, both in terms of regularity
and in terms of asymptotic behavior.
It is quite exceptional to find an explicit formula for the solution of
PDEs, hence approximation techniques are important, which
leads to discuss compactness and continuity issues.
Moreover, for many evolution problems of interest, the solution does not
preserve the regularity of the solutions, and we should work in a
framework of weak solutions.
The lectures will consider the following questions:
- Weak derivatives, weak solutions
- Functional spaces, in particular Sobolev spaces
- Linear problems, variational methods and spectral analysis.
- Classical PDEs: heat, wave, transport, Schrodinger...
- Introduction to non linear problems
