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.

## 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

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

## In the programs

• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Calculus of variations
• Lecture: 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
• Lecture: 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
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional

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