MATH-437 / 5 credits

Teacher:

Language: English

Remark: Cours donné en alternance tous les deux ans (pas donné en 2021-22)

Summary

Introduction to classical Calculus of Variations and a selection of modern techniques.

Keywords

calculus of variations, optimization, minimization, Euler-Lagrange equations, first variation, direct method, Lagrangian, functional analysis, Sobolev spaces, minimal surfaces, convexity, existence, uniqueness, regularity.

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:

• Illustrate historically important optimization problems
• Model geometrical and/or physical problems in the form of optimization
• Analyze the existence and uniqueness of minimizers of optimization problems
• Investigate the regularity properties of minimizers

Teaching methods

Lectures + 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

 Assistants Yes Forum No

No

Bibliography

Main reference:

• Introduction to the Calculus of Variations, B. Dacorogna

Other useful resources:

• Direct Methods in the Calculus of Variations, E. Giusti
• Introduction to the Modern Calculus of Variations, F. Rindler
• 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
• 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
• 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

Reference week

 Mo Tu We Th Fr 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