The course aims at providing an introduction to the modern theory of Calculus of Variations, which is a powerful tool to study many problems in mathematics, physics and applied sciences (for instsance: existence of geodesics, surfaces of minimal area, periodic solutions of N-body problems, existence of solutions for nonlinear elliptic PDE).
Expected learning outcomes
Acquisition of the basic notions and techniques in the theory of Calculus of Variations: minimization, deformations, problems of compactness, relations between topology and critical points. Study of the relations between critical point theory and partial differential equations.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
1.The direct method of the calculus of variations 2. The functional framework for differential equations 3. Minimax-theorems for indefinite functionals 4. Existence of solutions of nonlinear partial differential equations 5. Problems with symmetries and index theories 6. Problems with lack of compactness 7. Applications elliptic equations with nonlinearities with critical growth
Prerequisites for admission
- Real analysis - Partial differential equations
Recommended: - Functional analysis
Lectures in traditional mode, at the blackboard.
Ambrosetti, A., Malchiodi, A., Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007 Struwe, M., Variational Methods, Springer, 2000
Assessment methods and Criteria
The exam consists of a single oral exam (30 minutes) which serves to verify the theoretical knowledge acquired during the course and the ability to solve exercises similar to the one which were proposed during the course.