Calculus of variations

lmat2250  2022-2023  Louvain-la-Neuve

Calculus of variations
5.00 credits
30.0 h + 15.0 h

This biannual learning unit is being organized in 2022-2023
It is recommended that the student is familiar with the basic notions of the Lebesgue integral as covered in the LMAT1221 undergraduate course and of Functional Analysis as covered in the LMAT1321 undergraduate course.
Main themes
Direct method of the calculus of variations, minimax methods, symmetry properties of optimal solutions.
The theme of the course will vary.
Learning outcomes

At the end of this learning unit, the student is able to :

1 Contribution of the course to learning outcomes in the Master in Mathematics programme.
By the end of this activity, students will have made progress in: 
- Recognise the fundamental concepts of important current mathematical theories.
- Establish the main connections between these theories, analyse them and explain them.
- Recognise the fundamental concepts of important current mathematical theories.
- Identify the unifying aspects of different situations and experiences.
- Argue within the context of the axiomatic method.
- Construct and draw up a proof independently, clearly and rigorously.
- Write a mathematical text in French according to the conventions of the discipline.
- Structure an oral presentation and adapt it to the listeners' level of understanding.
- Find sources in the mathematical literature and assess their relevance.
- Correctly locate an advanced mathematical text in relation to knowledge acquired.
- Ask himself relevant and lucid questions on a mathematical topic in an independent manner.
Learning outcomes specific to the course (en fonction des thèmes traités).
Initiate to the current research in minima and critical points of integral functionals.
- Start a research project with a deeper knowledge of an area of current mathematics.

In particular, he will have developed his ability to:
-- Independently develop mathematical intuition by anticipating expected results (formulating conjectures) and checking for consistency with already existing results.
-- Independently ask relevant and lucid questions on an advanced mathematical topic.
The course will cover elements of Calculus of Variations in one variable :
  • optimization problems;
  • Euler-Lagrange equation;
  • absolutely continuous functions and one-dimensional Sobolev spaces;
  • not smooth extrema;
  • Ekeland's variational principle;
  • Mountain pass theorem.
Teaching methods
The learning activities consist of lectures and practical work sessions.
Lectures aim to introduce fundamental concepts, to motivate them by showing examples and establishing results, to show their reciprocal links and their links with other courses.
The practical work sessions aim to deepen the concepts covered during the lectures.
Evaluation methods
The acquisition of skills will be assessed during projects during the semester and a final exam.
The questions will ask :
  • restore material, including definitions, theorems, proofs, examples,
  • choose and apply methods of the course to solve problems and exercises,
  • adapt course demonstration methods to new situations,
  • synthesize and compare objects and concepts.
The evaluation will focus on :
  • knowledge, understanding and application of the different mathematical objects and methods of the course,
  • the rigor of the developments, proofs and justifications,
  • the quality of the writing of the responses.
Other information
This course will not be offered in 2021-2022.
Online resources
Additional documents on Moodle.
Le cours sera basé sur des extraits des références suivantes :
The course will be based on excerpts from the following references :
  1. Troutman, John L. Variational calculus and optimal control. Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996.
  2. Brezis, Haïm Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
  3. Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan One-dimensional variational problems. An introduction. Oxford Lecture Series in Mathematics and its Applications, 15. The Clarendon Press, Oxford University Press, New York, 1998.
  4. Clarke, Francis Functional analysis, calculus of variations and optimal control. Graduate Texts in Mathematics, 264. Springer, London, 2013.
  5. Mawhin, Jean; Willem, Michel Critical point theory and Hamiltonian systems. Applied Mathematical Sciences, 74. Springer-Verlag, New York, 1989.
Faculty or entity

Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme
Learning outcomes
Master [60] in Physics

Master [120] in Mathematics

Master [60] in Mathematics

Master [120] in Physics