96771 - Calculus of Variations

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

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 into geometric problems in Calculus of Variations and some of their developements in PDEs, and partly, in regularity of PDEs. In particular,

• in the first part it will be discussed some fine properties of sets of finite perimeter. This will be discussed as an invitation to the topic and will be held mostly without proofs.
• There will be proposed a slightly simplified but complete solution for the isoperimetric problem, stating that a ball minimizes the perimeter under volume constraint. The proof will follow mostly 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 those compact embedding 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 about rearrangement functional inequalities (we will see and partly prove Hardy-Littlewood, Riesz and Polya-Szego ineuqalities).
• As a third and last inequality, we will introduce some basics of Shape Optimization Theory. We will in particular show the Saint-Venant and the Faber-Krahn inequality.
• The last part of the course will deal about regularity. In particular we will offer a complete proof of the solution of the XIX Hilbert problem, via the original De Giorgi proof in 1956.

 

 

 

 

 

Readings/Bibliography

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.

Teaching methods

Frontal lectures

Assessment methods

Oral exam

Teaching tools

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

Office hours

See the website of Eleonora Cinti

See the website of Berardo Ruffini