CALCULUS OF VARIATIONS
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- Versione italiana
- Academic year
- 2015/2016
- Teacher
- MICHELE MIRANDA
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/05
Training objectives
- The main target of the course is the learning of some fundamental tool of the Calculus of Variations. In particular, we shall show the direct methods of the Calculus of Variations and develop the main tools for their application.
Then, a preliminary target shall be the study of the notion of compactness in infinite dimensional functional spaces, with the study of weak topologies
in suitable functional spaces. Finally, a last target of the course is the study of differential equations, both ODE (in the case of one-dimensional
calculus of variations, such as the equations of the geodesic and brachistocrona) and PDE (in the multi-dimensional case, such as Laplace equation and general elliptic equations).
Obviously, it is clear that the abilities that the students have to acquire during the course is the ability of recognize problems that can be treated in the variational setting, understanding whether or not the solutions of given problems exist and which regularity can be expected. With the methods exposed during the course, the students have to be able to determine
the solutions of such problems, at least in some particular cases. Prerequisites
- Elements of functional analysis; Hilbert spaces, Banach spaces, duality, weak topologies and strong and weak compactness theorems. Basic measure theory (also only Lebesgue measure); Lebesgue spaces and possibly Sobolev spaces.
Course programme
- The course is intended as an introduction to the direct methods in Calculus of Variations, with particular attention to the determination of minima of functionals defined on Sobolev spaces and on their properties.
In the first part of the course we introduce the notion of lower semicontinuity in topological and metrizable spaces; beside such
the notion of lower semicontinuous envelope is introduced. (6 hours)
We then introduce the Sobolev Spaces with the description of their
main properties; in particular, the continuous and compact embedding
of Sobolev Spaces in some functional spaces, such as spaces of continous and Hoelder continuous functions or suitable Lebesgue spaces. (10 hours)
At the end of this introductory material on Sobolev Spaces, we study lower semicontinuity properties of Lagrangian, showing how the notion of convexity is a necessary and sufficient condition in some suitable sense.
We then shall show existence and uniqueness theorems for minima of Lagrangian defined on Sobolev Spaces, showing how their determination can be performed by looking to the solutions of the associated Euler-
Lagrange equation. (10 hours)
In the last part of the course we shall study regularity properties of the minima; a first regularity study is the Sobolev one, whether a second type is linked with Hoelder regularity. Fundamental tools in such direction are givegn by the Caccioppoli and Harnack inequalities. (16 hours) Didactic methods
- The course is mainly based on theoretical lectures in which the main tools shall be presented; during the lectures there shall also be presented a more practial part, showing some applications of the theory.
Learning assessment procedures
- The verification of the acquired knowledges is based on a colloquium. The modality of such colloquium can be chosen by the student among the following.
1. Classical oral exam, in which the candidate has to answer to three questions on three different arguments treated during the course.
2. Presentation of two medium-small deepening that have to be chosen with the teacher on two different arguments treated during the course
3. Preparation of a seminar on an argument to be chosen with the teacher in which the student shows with detalis an argument, showing also the results used in the exposition.
The common characteristic of the three modalities is the duration of the colloquium, that has to be more or less 30-40 minutes. Reference texts
- As reference text we suggest;
Brezis,
"Functional Analysis, Sobolev Spaces and Partial Differential Equations",
Universitext, Springer
To deepen the arguments, we suggest also the following texts:
Buttazzo-Giaquinta-Hildebrandt,
"One-dimensional Variational Problems",
Oxford Lecture Series in Mathematics and its Applications
Giusti,
"Direct methods in the Calculus of Variations",
World Scientific Pub Co Inc
