81657 - ANALISI MATEMATICA 3

Learning outcomes

At he end of the lessons, the student knows some advanced modern mathematical analysis instruments: Hilbert and Banach spaces as well a some linear operators between these spaces, weak convergence, L^p spaces, Fourier transform in L^2,  Ascoli-Arzelà theorem. The student knows how to use this knowledge for facing and solving applied problems. He or She is habit to learn and is endowed of some specific skills for attending to the second order degree lessons.

Course contents

1) L^p spaces: completeness, density of some set of functions (simple functions, continuous functions), regularization (Friedrichs mollificators).

2) Fourier transform in L^1, in Schwarz space S and in L^2. Introduction to the temperate space distribution S' and to the distribution space D'. Basic to Laplace transform. 

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

4) Banach spaces, Hilbert spaces. Baire's theorem. Hahn-Banach theorem, Banach-Steinhaus theorem and closed graph theorem. Weak convergrnce and minima of coercive functionals and weakly lower semicontinuous functionals.

5) Basic abstrac measure theory. Borel and Radon measures. Lebesgue decomposition, Radon-Nikodym Theorem.

 

 

Readings/Bibliography

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

Teaching methods

Classical lessons.

Assessment methods

Didactic targets:

-to be able to introduce some topics discussed during the course, proving to have deeply understood the fundamental theoretical concepts;

-to be able of solving some exercises.

The exam is subdivided in a 30 minutes admission trial and written exam followed by a colloquium. The admission exam takes place in a different day with respect to the successive one hour written exam and subsequent colloquium. In order to be admetted to the one hour written exam and the colloqium as well, the candidate has to obtain at least 6/10 in the admission exam.

The final score is determined take in account of the admission exam as well as the one hour exam and successive colloqium as well as of the comprehension of the exercises assigned during the lessons.

Teaching tools

The lecture notes will be available on Virtuale as well as on IOL.

Links to further information

https://www.unibo.it/sitoweb/fausto.ferrari

Office hours

See the website of Fausto Ferrari