VARIATIONAL MEYHODS IN ANALYSIS
The course aims at introducing to some of the fundamentals topics and techniques of the Calculus of Variations and to their applications to other branches of Mathematical Analysis. Classes will be aimed at stimulating a deeper understanding of the ideas and methods of mathematical thinking. The main goals are: 1) improving the skills of the student in learning new mathematical tools; 2) improving the ability of the students to apply the newly learned theoretical notions in the solution of problems and exercises;
3) improving to student's skill in communicating with clarity and precision the mathematical concepts acquired during the course.
Measure Theory and basic Functional Analysis.
-Direct Methods in the Calculus of Variations and their application to Partial Differential Equations (PDE).
-Introduction to Regularity Theory: Campanato's approach, De Giorgi-Nash Theorem and their application to the study of the regularity of minimizers of some integral functionals of the Calculus of Variations.
-Introduction to Critical Point Theory: the Mountain Pass Theorem and its applications to PDE.
-Introduction to Gamma-convergence, examples and applications.
-Evolutions problems:the Minimizing Movements Method.
Complements (as time permits):
- Vectorial Calculus of Variations.
-Examples of functionals defined on spaces of discontinuous functions.
-"Partial Differential Equations" by L. C. Evans
-"An introduction to Gamma-convergence" byi G. Dal Maso
-"Gamma-convergence for beginners" by A. Braides
Lectures at the blackboard