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B0316 - ELEMENTI DI CALCOLO DELLE VARIAZIONI

Academic Year 2024/2025

                
                        Docente:
                        Giovanni Cupini
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/05
                    
                        Language:
                        Italian
                    
                        Moduli:
                        
                            Giovanni Cupini
                            (Modulo 1)
                        
                            Annalisa Baldi
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    Traditional lectures (Modulo 1)
                                
                                    Traditional lectures (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            First cycle degree programme (L) in
                            Mathematics (cod. 8010)

                            Teaching resources on Virtuale
                        
                                        Course Timetable
                                    
from Feb 18, 2025 to May 29, 2025

Learning outcomes

At the end of the course, the student knows the basic notions of the calculus of variations. He/she knows how to apply the knowledge acquired to the understanding and resolution of model problems originating from the theory of differential equations, also relating to electrostatics, mechanics and materials science.

Course contents

Recall/preliminaries on the Sobolev spaces, Sobolev and Poincaré inequalities, Rellich-type theorems.

Some classic problems of the calculus of variations, e.g. the isoperimetric inequality and the brachistochrone problem.

Integral functionals of the calculus of variations: coercivity and lower semicontinuity.

Existence of minima: the classical method and the direct method.

BV spaces, perimeter and isoperimetric inequalities.

Short introduction to Gamma-convergence.

Readings/Bibliography

G. Buttazzo, M. Giaquinta, S. Hildebrandt: One-dimensional variational problems. An introduction, Oxford Lecture Series in Mathematics and its Applications, 15. The Clarendon Press, Oxford University Press, New York, 1998.

B. Dacorogna: Introduction to the Calculus of Variations, Imperial College Press

L. Evans, R.F. Gariepy: Measure theory and fine properties of functions. Revised edition. CRC Press, Boca Raton, FL, 2015.

G. Talenti, A. Colesanti, P. Salani: Un'introduzione al calcolo delle variazioni. Teoria ed esercizi, Unione Matematica Italiana, Bologna, 2016.

Teaching methods

Lessons in the classroom.

Assessment methods

The examination consists of an oral test. The student must show to know the concepts explained during the course (in particular definitions, theorems and their proofs) and how to connect them.

Office hours

See the website of Giovanni Cupini

See the website of Annalisa Baldi