96771 - CALCULUS OF VARIATIONS

Conoscenze e abilità da conseguire

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.

Contenuti

MODULUS I

In the first part of the course, we will introduce some basic notions and methodologies in Calculus of Variations:

• We will start by describing the - so called- Direct Method, for proving existence of minimizers of variational functionals, in some easy situations.
  
• Then, we will introduce the class of Lipschitz functions and their main properties and study a minimization problem in such a class. This will be done under some particular assumption on the domain and on the boundary datum (the "bounded slope condition") and existence will be achieved via the constructions of suitable "barriers". As a particular case, we will consider the area functional for graphs.
• After this , we will introduce the class of BV functions (functions of bounded variations) and the notion of set of finite perimeter. We will state and prove some of their main properties.
• This will allow us to consider geometric minimization problem and to establish an existence result for minimizers for a Plateau-type problem.
• Finally, we will define the notion of Reduced boundary of a set of finite perimeter and we will state the De Giorgi's structure Theorem.

MODULUS II

In the second part of the course, we will deal in a mixed way with geometric problems in the Calculus of Variations and some of their developments in PDEs, and partly, with the regularity of PDEs. In particular, in the first part, some fine properties of sets of finite perimeter will be discussed. This will be presented as an invitation to the topic and will be held mostly without proofs.

• A slightly simplified but complete solution for the isoperimetric problem will be proposed, stating that a ball minimizes perimeter under a volume constraint. The proof will mostly follow the basic De Giorgi approach, after the Steiner symmetrization technique.

• As a first application, we will show how to derive the (equivalent) Sobolev inequalities via isoperimetric ones. In particular, we will see how Sobolev inequalities are the functional counterpart of the geometric isoperimetric inequality, and how to derive, by means of these, compact embeddings of Sobolev spaces into integrability spaces.

• As a second application, we will see how mild (Sobolev) regularity for elliptic PDEs can be derived by means of Sobolev inequalities. This will be done via a short introduction to rearrangement functional inequalities (we will see and partly prove Hardy-Littlewood, Riesz, and Polya-Szego inequalities).

• As a third and last topic, we will introduce some basics of Shape Optimization Theory. We will, in particular, show the Saint-Venant and the Faber-Krahn inequalities.

• The last part of the course will deal with regularity. In particular, we will offer a complete proof of the solution to the XIX Hilbert problem, via the original De Giorgi proof from 1956.

Testi/Bibliografia

Luigi Ambrosio, Nicola Fusco, Diego Pallara, "Functions of Bounded Variation 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 Variation".

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

Metodi didattici

Frontal lectures

Modalità di verifica e valutazione dell'apprendimento

Oral exam

Strumenti a supporto della didattica

The suggested Textbooks. The notes of the whole course will be available on "Virtuale".

Orario di ricevimento

Consulta il sito web di Berardo Ruffini

Consulta il sito web di Eleonora Cinti