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In undergraduate calculus we learn to maximise or minimise functions of one variable, finding optimal points. The Calculus of Variations is concerned with finding optimal functions and the properties of these optimisers. Famous examples include surfaces of minimal area or the shortest or quickest paths between given points. In the classical indirect method, the optimisers are found as solutions of the Euler-Lagrange differential equations. In the modern direct method, one uses abstract means to find optimisers, which often yields existence results for solutions of differential equations.


Autumn 2019 (Monday, October 7 to Friday, December 13)


  • Live lecture hours: 10
  • Recorded lecture hours: 0
  • Total advised study hours: 40


  • Mon 14:05 - 14:55


Mostly undergraduate calculus, but we will also use the implicit function theorem and some existence and uniqueness theory for ODEs. Knowledge of Lebesgue integration is helpful but not required.


In this 10-lecture series, elements of both the classical and the modern theory are presented. A tentative plan is as follows:

    1. Introduction. One-dimensional variational problems: Fundamental lemma, Euler-Lagrange equations
    2.-3. Convexity and existence and regularity issues
    4.-5. Second variations and necessary conditions for optimality
    6. Variational problems with constraints
    7. Problems involving multiple integrals
    8. Direct method: coercivity and lower semicontinuity
    9. Sobolev spaces, weak convergence and compactness
    10. The direct method for integral functionals and existence of solutions for some nonlinear PDEs

Other courses that you may be interested in:


Matthias Kurzke
Phone 0115 9514984


Calculus of VariationsJost and Li-Jost
Calculus of Variations I: The Lagrangian FormalismGiaquinta and Hildebrandt
Introduction to the Calculus of VariationsSagan
Calculus of VariationsGelfand and Fomin
One-dimensional variational problemsButtazzo, Giaquinta and Hildebrandt


Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)


The assessment for this course will be via a single take-home paper in January with 2 weeks to complete and submit online. There will be 4 or 5 questions, each giving between 10 and 30 marks for a total of 100 marks, and you will need 50 marks to pass.

Calculus of Variations Exam 2019/20

Files:Exam paper
Released: Monday 6 January 2020 (184.6 days ago)
Deadline: Sunday 19 January 2020 (170.6 days ago)


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Recorded Lectures

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