- Docente: Eleonora Cinti
- Credits: 6
- SSD: MAT/05
- Language: English
- 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)
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.
Applications: the Newton problem and the Farber-Krahn inequality.
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