MAGIC095: Calculus of Variations

Course details

A specialist MAGIC course


Autumn 2017
Monday, October 9th to Friday, December 15th


Live lecture hours
Recorded lecture hours
Total advised study hours


13:05 - 13:55


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.


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


  • MK

    Dr Matthias Kurzke

    University of Nottingham
    Main contact
  • Yv

    Dr Yves van Gennip

    University of Nottingham


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The assessment for this course will be released on Monday 8th January 2018 and is due in by Sunday 21st January 2018 at 23:59.

The assessment for this course will be by a single take-home paper, to complete during the official assessment period in January. The paper will be organised in questions for which marks will be given. Half of the total marks will be required to pass the exam.

Please note that you are not registered for assessment on this course.


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