81657 - ANALISI MATEMATICA 3

Learning outcomes

At the end of the course, students should know modern advanced tools in mathematical analysis: Fourier transform, Hilbert and Banach spaces, abstract measure theory, weak derivatives. Student should be able to use these tools to attack and solve non-elementary problems in applied sciences, in particular those that can be modeled in terms of partial differential equations. Finally, they should reach learning skills and high level ok knowledge, making them able to accede to master lectures and programs.

Course contents

1) Elements of abstract measure theory. Borel measures, Radon measures. Lebesgue decomposition. Radon-Nikodym theorem.
L^p spaces: completeness, density of some classes of functions (simple functions, continuous functions), regularization (Friedrichs' mollifiers).

2) Banach and Hilbert spaces. Baire Theorem, Hahn-Banach, Banach-Steinhaus and closed graph Theorems. Weak convergence and minima of coercive weakly lower semicontinuous functionals.

3) Weak derivatives and Sobolev spaces. Lax-Milgram Theorem and Dirichlet problem for second order elliptic operators.

4) Fourier transform il L^1, in the Schwarz space S, and in L^2.

Readings/Bibliography

H. Brezis, Sobolev Spaces and Partial Differential Equations, Springer, New York.
W. Rudin, Analisi reale e complessa, Boringhieri, Torino

Teaching methods

Lectures at the blackboard

Assessment methods

The exam aims to verify the achievement of the following educational objectives: the student should show

- a coherent presentation of some topics of the course, proving to have fully understood the fundamental concepts and the deduction mechanisms;

- a possible resolution of exercises related to the topics covered.

The exam consists of a 2-hour written test (which will consist of a short dissertation on one of the course topics) and of an oral interview, which must both be taken in the same session. In order to take the oral exam, it is necessary to have obtained a score of not less than 18/30 in the written test.

The final grade takes into account the results achieved in both tests.

Office hours

See the website of Bruno Franchi