Calculus of variations
MATH-437 / 5 credits
Teacher: Michelat Alexis Paul Benjamin
Language: English
Remark: Cours donné en alternance tous les deux ans
Summary
Introduction to classical Calculus of Variations and a selection of modern techniques. The Calculus of Variations aims at showing the existence of minimisers (or critical points) of functionals that naturally appear in mathematics and physics (Dirichlet energy, p-energy, etc).
Content
- Preliminaries: weak convergence, Sobolev spaces;
- Classical methods: Euler-Lagrange equation and other necessary minimality conditions;
- Direct methods: coercivity, lower-semicontinuity, (quasi-)convexity, relaxation, Lavrentiev phenomenon;
- If time permits: Gamma-convergence.
Keywords
Calculus of variations; minimisation; integral functionals; Euler-Lagrange equations; variations; direct method of the calculus of variations; lower semi-continuity; Sobolev spaces; (quasi-)convexity; existence and uniqueness of minimisers.
Learning Prerequisites
Required courses
- MATH-200: Analysis III
- MATH-205: Analysis IV
- MATH-303: Measure and integration
Recommended courses
- MATH-301: Ordinary differential equations
- MATH-302: Functional analysis I
- MATH-305: Sobolev spaces and elliptic equations
Important concepts to start the course
The students are required to have sufficient knowledge on real analysis and measure theory. Having taken a course on functional analysis or Sobolev spaces will be an advantage.
Learning Outcomes
By the end of the course, the student must be able to:
- Discuss the assumptions in a minimization problem
- Apply the direct method of the calculus of variations
- Analyze the existence and uniqueness of minimizers of optimization problems
- Derive the Euler-Lagrange equation and other necessary conditions for minimizers
- Distinguish between scalar and vectorial minimization problems
Teaching methods
Lectures + exercises.
Expected student activities
Following the lectures and solving exercises
Assessment methods
Oral exam.
Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.
Supervision
Office hours | No |
Assistants | Yes |
Forum | Yes |
Resources
Virtual desktop infrastructure (VDI)
No
Bibliography
Main reference:
- Introduction to the Calculus of Variations, B. Dacorogna
Other useful resources:
- Weak Convergence Methods for Nonlinear Partial Differential Equations
L. C. Evans - Direct Methods in the Calculus of Variations, E. Giusti
- Functional Analysis, Sobolev Spaces and Partial Differential Equations, H. Brezis
- Partial Differential Equations, L. C. Evans
Ressources en bibliothèque
- Functional Analysis, Sobolev Spaces and Partial Differential Equations / H. Brezis
- Introduction to the Calculus of Variations / Dacorogna
- Partial Differential Equations / L. C. Evans
- Direct Methods in the Calculus of Variations /Giusti
Moodle Link
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Calculus of variations
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Calculus of variations
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Calculus of variations
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional