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96735 - Advanced Methods of Mathematical Analysis

Academic Year 2021/2022

                
                        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. 5827)

                                Also valid for
                                
                                    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 Sobolev spaces. 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. A. Parmeggiani): Sovability of partial differential operators acting on Hilbert spaces. Quick review of temperate distributions. Sobolev spaces on Rn, duality between Sobolev space and applications to regularity theory of PDEs. Sobolev spaces on a bounded open set and applications to the spectral theory of the Laplacian with homogeneous boundary data (homogeneous Dirichlet data). A priori estimates for first order systems that play a crucial role in the study of the d-bar operator on open subset of Cn.

Part 2 (prof. F. Ferrari): Variational inequalities. Variational inequalities in Hilbert spaces, with particular focus on Sobolev spaces and introductions to free boundary problems. Introduction to the theory of viscous solutions to nonlinear PDEs.

Readings/Bibliography

Part 1. 1) G. Grubb: Distributions and Operators, Graduate Texts in Mathematics 252, Springer 2) G. Folland: Introduction to Partial Differential Equations. Second Edition. Princeton University Press.

Part 2. 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)

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