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.

## 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

## 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

## 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
• Type: optional
• 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
• Type: optional
• 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
• Type: optional
• 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
• Type: optional

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