Roberta Marziani
University of Siena - Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche
Course Type
Type A
Calendar
April 7 h. 14:00-17:00 Aula E April 8 h. 14:00-17:00 Aula CD April 9 h. 14:00-17:00 Aula 18 April 11 h. 14:00-17:00 Aula 14 April 14 h. 9:30-12:30 Aula 101 April 15 h. 10:00- 12:00 Aula 14 April 16 h. 14:00-17:00 Aula 18
Room
Program
Brief abstract
The calculus of variations is a field of mathematics concerned with minimizing (or maximizing) a certain quantity (usually a functional) within a given class of objects (usually functions). It has a wide range of applications in physics, engineering, applied and pure mathematics, and is
intimately connected to partial differential equations (PDEs).
The aim of the course is to present the basic notions and some of the classical results of this
field with as less prerequisites as possible. This will allow to consider some applications to
classical mechanics and materials science.
Syllabus:
• Examples of minimization problems in one dimension: the brachistochrone, the Fermat problem, the catenary problem, rotationally symmetric minimal surfaces;
• Example of a functional without minimizers;
• Classical methods in the Calculus of Variations: fundamental lemma, the Euler-Lagrange equations, minima versus critical points;
• First comments on the regularity of minimizers;
• More on rotationally symmetric minimal surfaces;
• Some hint of Γ-convergence, application to phase transition and image segmentation problems.
