# Topics in calculus of variations

MATH-515 / **5 credits**

**Teacher: **

**Language:** English

**Remark:** pas donné en 2023-24

## Summary

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

## Content

- Classic functionals in the Calculus of Variations
- Semi-direct methods
- Direct method in Calculus of Variations
- Functionals in Sobolev spaces, convexity, lower semicontinuity, existence and regularity
- If time allows: Plateau's problem, Gamma-convergence, isoperimetric problem

## Keywords

calculus of variations, optimization, minimization, Euler-Lagrange equations, first variation, direct method, Lagrangian, convexity, lower semicontinuity.

## 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
- MATH-437: Calculus of Variatinos

## Learning Outcomes

By the end of the course, the student must be able to:

- Demonstrate proficiency in statements
- Identify use and role of the assumptions
- Recognize which concepts and results could be used in a given context
- Describe concepts and proofs
- Apply theory for specific examples

## Teaching methods

Lectures + Exercises

## Assessment methods

Oral

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 |

## Resources

## Bibliography

"*Introduction to the Calculus of Variations*", B. Dacorogna

"*Direct Methods in Calculus of Variations*", B. Dacorogna

"*Calculus of Variations*", J. Jost & X. Li-Jost

"*One-dimensional Variational Problems*", G. Buttazzo & M. Giaquinta & S. Hildebrandt

*"Introduction to the Modern Calculus of Variations*", F. Rindler

"*Sets of Finite Perimeter and Geometric Variational Prob- lems: An Introduction to Geometric Measure Theory*", F. Maggi

"*Measure Theory and Fine Properties of Functions*", L.C. Evans & R.F. Gariepy

## Ressources en bibliothèque

- One-dimensional Variational Problems / Buttazzo
- Introduction to the Calculus of Variations / Dacorogna
- Calculus of Variations / Jost
- Direct Methods in Calculus of Variations / Dacorogna
- Introduction to the Modern Calculus of Variations / Rindler
- Sets of Finite Perimeter and Geometric Variational Prob- lems: An Introduction to Geometric Measure Theory / Maggi
- Measure Theory and Fine Properties of Functions / Evans

## In the programs

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

## Reference week

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