# Coursebooks

## Calculus of variations

Stra Federico

English

#### Summary

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

#### Content

• Preliminaries: Hölder functions, Sobolev spaces, functional analysis, convex analysis...
• Model problems: geodesics, brachistochrone, minimal surfaces, isoperimetric problem, Lagrangian mechanics...
• Classical methods: Euler-Lagrange equation, first and second variations...
• Direct methods: coercivity, compactness, lower-semicontinuity...
• Regularity theory for minimizers: Sobolev regularity, Hölder regularity
• Optional if time permits: isoperimetric inequality, '-convergence

#### Keywords

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

#### 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:
• 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

#### Resources

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

### Reference week

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Under construction

Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German