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34686 - Fundamentals of Advanced Analysis 2

Academic Year 2019/2020

                
                        Docente:
                        Alberto Parmeggiani
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/05
                    
                        Language:
                        Italian
                    
                        Moduli:
                        
                            Alberto Parmeggiani
                            (Modulo 1)
                        
                            Fausto Ferrari
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    Traditional lectures (Modulo 1)
                                
                                    Traditional lectures (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            Second cycle degree programme (LM) in
                            Mathematics (cod. 8208)

                            Teaching resources on Virtuale

Learning outcomes

The course aims at giving the student the fundamentals of the theory of distributions, Sobolev spaces and abstract measure theory. At the end of the course the student will be able to independently study abstract and applied theories that require the knowledge of the aforementioned theories.

Course contents

The course is organized in two parts. Part 1 (prof. F. Ferrari) Variational inequalities, Variational inequalities in Hilbert spaces with particular attention to Sobolev spaces and the obstacle problem. Part 2 (prof. A. Parmeggiani) Schwarz class of rapidly decreasing functions. Temperate distributions. Fourier transform of Schwartz functions and temperate distributions. Applications to partial differential equations.

Readings/Bibliography

Part 1. Some of the topics will be discussed following the following books: D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications Academic Press 1980, R. Adams, Sobolev Spaces Academic Press 1975. In addition might be useful also: L.C. Evans, R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Flo. (1992). W. Ziemer, Weakly differentiable functions: Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120, Springer-Verlag, New York (1989) Part 2. 1) L. Grafakos: Classical Fourier Analysis, Graduate Texts in Mathematics 249, 2nd Edition, Springer 2) G. Eskin: Lectures on Linear Partial Differential Equations, Graduate Studies in Mathematics Vol. 123, American Mathematical Society Further reading: 3) G. Grubb: Distributions and Operators, Graduate Texts in Mathematics 252, Springer 4) J. Rauch: Partial Differential Equations, Graduate Texts in Mathematics 128, Springer-Verlag 5) G. Folland: Introduction to Partial Differential Equations. Second Edition. Princeton University Press.

Teaching methods

Lectures at the blackboard and advanced seminars of the students.

Assessment methods

Students may choose between a traditional oral exam and a 45 minutes seminar on an advanced topic related to the topics of the course that has not been developed within the course itself. Some exercises will be proposed to the students during the course and the discussion of their solutions will be part of the final exam.

Office hours

See the website of Alberto Parmeggiani

See the website of Fausto Ferrari