-H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equation, Springer-Verlag New York, Universitext series, 2011
- K. Yosida, Functional Analysis, Springer-Verlag, Berlin-Heidelberg-New York 1966
- L. C. Evans, Partial Differential Equations, AMS, Vol.19
- Elvira Mascolo, Appunti di Analisi Funzionale,(Notes in Italian) can be downloaded at http://web.math.unifi.it/users/mascolo/DIDATTICA-MATEMATICA/libroAf.pdf
Learning Objectives
The course aims to provide basic knowledge of the main results related to linear operators, weak convergence, Sobolev spaces and variational methods for the study of differential equations.
Each topic will be complemented with examples and exercises, to allow the acquisition of a correct deductive method. At the end of the course students must be able to correctly perform exercises related to the proposed topics and may, for example, deal with problems of Calculus of Variations and Elliptic Partial Differential Equations with appropriate analytical tools.
Prerequisites
Lebesgue measure theory and L^p spaces.
Prerequisites: Content of the course Analisi Matematica 3
Suggested: Content of the course of Istituzioni di Analisi Superiore.
Teaching Methods
Frontal lectures: the content of the course will be presented in an analytical way.
Student-Teacher interaction will be strongly encouraged in order to facilitate a full understanding of the subject.
Type of Assessment
During the exam,which will be only oral, some questions will be posed both of a theoretical nature and about the resolution of problems.
More precisely, the student will be asked to prove some theorems among those presented during the course ( a list of theorem will be given), and to carry out some exercises. The aim is to verify the knowledge and degree of understanding of the theory developed in the course. The ability to communicate the subject in an analytical way and the use of an appropriate mathematical language will also be evaluated with particular attention.
Course program
The Hahn-Banach Theorem. The closed graph theorem. The open map theorem.
Duality, weak topologies. Separable and reflexive spaces.
Hilbert Spaces: projections on convex and closed sets, duality. Lax-Milgram theorem, orthonormal systems.
Compact operators: Riesz-Fredholm theory, spectral theory for compact, self-adjoint operators.
Sobolev Spaces: weak derivatives a main properties for first order Sobolev spaces. Variational formulation of uniformly elliptic partial differential equations. Fixed point theorems and Applications.