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96771 - Calculus of Variations

Academic Year 2021/2022

                
                        Docente:
                        Eleonora Cinti
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/05
                    
                        Language:
                        Italian
                    
                        Moduli:
                        
                            Eleonora Cinti
                            (Modulo 1)
                        
                            Berardo Ruffini
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    Traditional lectures (Modulo 1)
                                
                                    Traditional lectures (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            Second cycle degree programme (LM) in
                            Mathematics (cod. 5827)

                            Teaching resources on Virtuale

Learning outcomes

At the end of the course, the student has a knowledge of some advanced chapters of classical and direct methods in calculus variations, with application to some topic of deep recent interest.

Course contents

Existence of minimizers in the class of Lipschitz functions.

The area functional.

BV functions and their main properties.

Sets of finite perimeter and their main properties.

Existence of minimizers for geometric problems in the class of sets of finite perimeter.

Reduced boundary and De Giorgi's structure Theorem.

The isoperimetric problem and related topics

Rearrangement inequalities. The Pòlya-Szego inequality, the Hardy-Riesz-Sobolev inequality and some applications.

An introduction to Gamma-convergence, the Modica-Mortola Theorem.

Readings/Bibliography

Luigi Ambrosio, Nicola Fusco, Diego Pallara, "Functions of Bounded Variations and Free Discontinuity Problems.

L.C. Evans, L. F Gariepy, "Measure Theory and Fine Properties of Functions".

Enrico Giusti, "Direct Methods in the Calculus of Variations".

Enrico Giusti, "Minimal Surfaces and Functions of Bounded Variations".

Francesco Maggi, "Sets of Finite Perimeter and Geometric Variationsl Problems.

Teaching methods

Frontal lectures

Assessment methods

Oral exam

Teaching tools

The suggested Textbooks.

Office hours

See the website of Eleonora Cinti

See the website of Berardo Ruffini