Calculus of Variations
A.Y. 2024/2025
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 can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
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.
Te Nemitsky operator between L^p spaces. A brief introduction to Sobolev Spaces. Sobolev embeddings: analysis of the non compact case. Optimization in Banach Spaces. (Weak) lower semcontinuous functions. Existence of mimima of coercive functionals on weakly closed set. The role of convexity. 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. The topological degree. The generalized Mountain Pass Theorem. The topological linking. Applications to PDEs
- Problems with lack of compactness: elliptic problems with critical growth. The sharp Sobolev embedding constant. The symmetric decreasing rearrangement. The Pohozaev identity. The Brezis-Nirenberg Result.
- Even functionals: the Krasnoselskii index. The Symmetric Mountain Pass Theorem.
- The direct method of Calculus of Variations: some elements of functional analysis. Gateaux and Frechet derivatives.
Te Nemitsky operator between L^p spaces. A brief introduction to Sobolev Spaces. Sobolev embeddings: analysis of the non compact case. Optimization in Banach Spaces. (Weak) lower semcontinuous functions. Existence of mimima of coercive functionals on weakly closed set. The role of convexity. 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. The topological degree. The generalized Mountain Pass Theorem. The topological linking. Applications to PDEs
- Problems with lack of compactness: elliptic problems with critical growth. The sharp Sobolev embedding constant. The symmetric decreasing rearrangement. The Pohozaev identity. The Brezis-Nirenberg Result.
- Even functionals: the Krasnoselskii index. The Symmetric Mountain Pass Theorem.
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.
P. Rabinowitz, Minimax Methods in Critical Point Theory with applications to differential equations, American Mathematical Society n.65.
Eventual notes on the Ariel web page of the course.
Struwe, M., Variational Methods, Springer, 2000.
P. Rabinowitz, Minimax Methods in Critical Point Theory with applications to differential equations, American Mathematical Society n.65.
Eventual notes on the Ariel web page of the course.
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.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Tarsi Cristina
Shifts:
Turno
Professor:
Tarsi CristinaProfessor(s)