- 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
- E. Mascolo, Appunti di Analisi Funzionale,(Notes in Italian) 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.
Mandatory courses: Analisi Matematica III
Suggested courses: 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
The examination consists of a written and an oral test. The exercises in the written examination are assigned a few weeks in advance of the oral examination. During the latter, some theoretical questions assigned in advance will be asked, and possibly some problems will be solved, and the exercises of the written proof will be discussed.
More specifically, the students will be asked to demonstrate a number of theorems from those developed during the course (indicated on the syllabus), and to perform exercises to test their knowledge and degree of understanding of the theory developed in the course. Particular attention will also be paid to both the ability to communicate the subject matter critically and the use of appropriate mathematical language.
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.