The goal of the course is to give an introduction to Calculus of Variations. In particular the students, after having attended the course and passed the exam, will have a good knowledge of classical topics of the calculus of variations and of some more modern developments of the theory.
Course Prerequisites
All the courses of Analysis of the Bachelor and Functional Analysis. Basic knowledges of functional analysis and measure theory. The main results needed will be recalled during the lectures.
Teaching Methods
Lectures
Assessment Methods
Oral exam on the topics of the couse. This exam is aimed at verifying the degree of understanding of the theoretical topics covered in class, the clarity of presentation but also the ability to apply these notions in applied situations. For this reason the student will be required to have a substantial understanding of all the theory presented which can be verified both through questions on specific topics and through the proposal of problems related to the topics of the course and which can be solved using the tools introduced during the lessons. The questions will be structured on variable difficulty in order to establish the degree of depth in the acquisition of these skills. The formulation of the grade will be obtained by evaluating the overall breadth and depth of learning, as well as the clarity of the presentation and the skills demonstrated in problem solving.
Texts
B. Dacorogna, "Introduction to the Calculus of Variations", Imperial College Press 1992.
B. Dacorogna, "Direct Methods in the Calculus of Variations", Springer 2007.
G. Buttazzo, M. Giaquinta, S. Hildebrandt, “One-dimensional Variational Problems, an Introduction”, Oxford University Press, 1998.
E. Giusti, “Direct Methods in the Calculus of Variations”, World Scientific 2003.
D. Bucur, G. Buttazzo, "Variational Methods in Shape Optimization Problems", Birkhauser 2005.
G. Allaire, Shape optimization by the homogenization method, Springer-Verlag, 2002.
H. Attouch, Variational Convergence for Functions and Operators, Pitman, 1984.
A. Braides, "Gamma-convergence for Beginners", Oxford University Press 2002.
Contents
First part: Direct method of the Calculus of Variations, existence results , necessary conditions, sufficient conditions, regularity for integral functionals. Functional setting, classical and weak. Semicontinuity and convexity. Relaxation. Lavrentiev phenomenon. Notable examples (Dirichlet energy, Brachistochrone, Area functional...)
Second part: Homogenisation. H-convergence and its properties. Compactness of H-convergence. Div-Curl Lemma. G-convergence and its properties. Gamma-convergence and its properties. Fundamental Theorem of Gamma-convergence. Perturbation with continuous functions. Homogenisation of periodic symmetric coefficients. Equivalence between G-convergence and Gamma-convergence. Other examples: "cloud of ice" and Neumann sieve.
Course Language
Italian
More information
Students who are in one of the categories identified by "progetto sulla didattica innovativa" will be able to do office hours also online or at hours to be chosen with the lecturer, and to have available the notes of the lecturer.
