Calculus of Variations
A.Y. 2023/2024
Learning objectives
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
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Live classes will be given by using Zoom.us. Videos with lessons will be eventually made available at the official web space of this course, according to the instructions given by the university, and they will cover the topic of each week. The details of these meetings will be available in the course web page of the Ariel platform, as well as the videos and every kind of material needed for the course.
The exams will be done by following the ways suggested on the web page of the university. The exams will have the same structure as the ones done in presence.
The exams will be done by following the ways suggested on the web page of the university. The exams will have the same structure as the ones done in presence.
Course syllabus
- Introduction to the Calculus of Variations: hystorical notes. The Brachistochrone problem. The counterexample by Weierstass.
- The direct method of Calculus of Variations: some elements of functional analysis. Gateaux and Frechet derivatives.
The Euler equation. A brief introduction to Sobolev Spaces. Optimization in Banach Spaces. (Weak) lower semcontinuous functions. Existence of mimima of coercive functionals on weakly closed set. Applications to PDEs.
- Elliptic second order differential operators. Constrained optimization. The implicit function theorem. The energy functional and the lagrangian multipliers. The first eigenvalue of the Laplace operator. Applications to elliptic PDEs with subcritical polynomial growth.
- Minimax Theorems. Critical points of functionals and toplogy of the sublevel sets. The Deformation Lemma. The Palais-Smale condition. Pseudogradients. The Mountain Pass Theorem and applications to nonlinear PDEs.
- Problems in presence of symmetry. The topological degree and the Leray-Schauder degree. The topological linking.
Even functionals: the Ljusternik-Schnirelmann theory. The Symmetric Mountain Pass Theorem.
- Problems with lack of compactness: elliptic problems with critical growth. The sharp Sobolev embedding constant. The Pohozaev identity. The Brezis-Nirenberg Result.
- The direct method of Calculus of Variations: some elements of functional analysis. Gateaux and Frechet derivatives.
The Euler equation. A brief introduction to Sobolev Spaces. Optimization in Banach Spaces. (Weak) lower semcontinuous functions. Existence of mimima of coercive functionals on weakly closed set. Applications to PDEs.
- Elliptic second order differential operators. Constrained optimization. The implicit function theorem. The energy functional and the lagrangian multipliers. The first eigenvalue of the Laplace operator. Applications to elliptic PDEs with subcritical polynomial growth.
- Minimax Theorems. Critical points of functionals and toplogy of the sublevel sets. The Deformation Lemma. The Palais-Smale condition. Pseudogradients. The Mountain Pass Theorem and applications to nonlinear PDEs.
- Problems in presence of symmetry. The topological degree and the Leray-Schauder degree. The topological linking.
Even functionals: the Ljusternik-Schnirelmann theory. The Symmetric Mountain Pass Theorem.
- Problems with lack of compactness: elliptic problems with critical growth. The sharp Sobolev embedding constant. The Pohozaev identity. The Brezis-Nirenberg Result.
Prerequisites for admission
Topics of Real Anlysis and Partial Differential Equations.
Suggested: Functional Analysis.
Suggested: Functional Analysis.
Teaching methods
Lectures in traditional mode.
Teaching Resources
Ambrosetti, A., Malchiodi, A., Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007
Struwe, M., Variational Methods, Springer, 2000
Struwe, M., Variational Methods, Springer, 2000
Assessment methods and Criteria
The exam consists of a single oral exam (about 45 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. The candidate will be requested to state and prove some theorems , as well as to summarize them in a more general framework.
Educational website(s)
Professor(s)