CALCULUS OF VARIATIONS
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- Versione italiana
- Academic year
- 2021/2022
- Teacher
- LORENZO BRASCO
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/05
Training objectives
- The course is intended as an introduction to the realm of the Calculus of Variations. We will start with a tour of the classical one-dimensional variational problems, then we will pass to study the so-called Direct Methods in the Calculus of Variations, up to arrive (time permitting) to the De Giorgi-Nash-Moser Theorem, which marks the beginning of the modern Regularity Theory for solutions of problems in the Calculus of Variations. En passant, we will show how some purely geometric problems (i.e. the isoperimetric problem) can be formulated, tackled and solved through the analytical tools of the Calculus of Variations.
The first target will be to understand the link between elliptic partial differential equations (like the Laplace's equation or the minimal surfaces equation) and convex optimization problems. Then, we will face the problem of how to show existence of a solution, through a suitable infinite-dimensional generalization of the Weierstrass' Theorem. The final target will be to learn the first ideas and tools of regularity techniques, needed to show that the solutions found through the Direct Methods, have the desired regularity properties.
At the end of the course, the student must be able to recognize problems that can be handled through variational methods: in particular, he/she must be able to understand whether solutions to these problems exist or not and what should be their expect regularity. Prerequisites
- All the contents of the courses Mathematical Analysis I & II. Lebesgue measure and integration. Theory of L^p 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.
TOOLS (8 hours)
- convex functions of one variable
- convex functions of several variables
- Picone's inequality
- Young's inequality
- Jensen's inequality
- the Du Bois-Reymond's lemma
- the Dirichlet's principle
- first variation
- Euler-Lagrange equation in weak form
SOME ONE-DIMENSIONAL VARIATIONAL PROBLEMS (12 hours)
- curves of minimal length
- curves of minimal kinetic energy
- the brachistocrone problem
- Poincaré's inequality on an interval
- finding the sharp constants: functions vanishing at the boundary, functions with vanishing mean, periodic functions with vanishing mean
- isoperimetric problems in the plane
- Hurwitz's isoperimetric inequality
SOBOLEV SPACES (20 hours)
- some motivations: the Direct Method
- weak derivatives in L^p
- definition of Sobolev space
- approximation of Sobolev functions by convolutions
- operations on Sobolev functions
- Poincaré inequality
- Sobolev inequality
- Ladyzhenskaya inequality
- Morrey inequality
- the space W^{1,p}_0
- embedding theorems for W^{1,p}_0
- embedding theorems for W^{1,p}
- counterexamples to the embedding theorems for rough domains
THE DIRECT METHODS ON SOBOLEV SPACES (8 hours)
- a model case: the Poisson equation
- an existence result in Sobolev spaces
- the first eigenvalue of the Dirichlet-Laplacian
- the spectrum of the Dirichlet-Laplacian
- excerpts of Regularity Theory: eigenfunctions are essentially bounded 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 an oral examination, where the student is asked to answer 2 or 3 questions on different arguments treated during the course. The examination lasts one hour, approximatively.
Reference texts
- The course will be based on the book (in english) written by the teacher
L. Brasco, "Handbook of Calculus of Variations for absolute beginners"
For the prerequisites on L^p spaces, we recommend
E. H Lieb, M. Loss, "Analysis", Graduate Texts in Mathematics, AMS (Chapter 2)
Students interested in deepen the arguments may consult the following texts:
H. Brezis,
"Functional Analysis, Sobolev Spaces and Partial Differential Equations",
Universitext, Springer
B. Dacorogna, "Introduction to the Calculus of Variations"
E. Giusti,
"Direct methods in the Calculus of Variations",
World Scientific Pub Co Inc
G. Talenti, A. Colesanti, P. Salani, "Un'introduzione al Calcolo delle Variazioni", Unione Matematica Italiana
