MATH-437 / 5 crédits

Enseignant: Ruf Matthias Benjamin

Langue: Anglais

Remark: donné en alternance tous les deux ans

## Summary

Introduction to classical Calculus of Variations and a selection of modern techniques. We focus on inegral functionals defined on Sobolev spaces.

## Keywords

calculus of variations, minimization, integral functionals, Euler-Lagrange equations, variations, direct method, lower semicontinuity, Sobolev spaces, (quasi-)convexity, existence and uniqueness of minimizers.

## 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:

• 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

## Dans les plans d'études

• Semestre: Automne
• Forme de l'examen: Oral (session d'hiver)
• Matière examinée: Calculus of variations
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Semestre: Automne
• Forme de l'examen: Oral (session d'hiver)
• Matière examinée: Calculus of variations
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Semestre: Automne
• Forme de l'examen: Oral (session d'hiver)
• Matière examinée: Calculus of variations
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Semestre: Automne
• Forme de l'examen: Oral (session d'hiver)
• Matière examinée: Calculus of variations
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines

## Semaine de référence

 Lu Ma Me Je Ve 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-21 21-22

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