MATH-437 / 5 credits

Teacher: Ruf Matthias Benjamin

Language: English

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

## In the programs

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

## Reference week

 Mo Tu We Th Fr 8-9 9-10 10-11 11-12 12-13 13-14 14-15 15-16 MAA330 16-17 17-18 MAA330 18-19 19-20 20-21 21-22

Tuesday, 15h - 17h: Lecture MAA330

Tuesday, 17h - 19h: Exercise, TP MAA330