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General


Description

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.

Semester

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

Hours

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

Timetable

  • Mon 14:05 - 14:55

Prerequisites

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.

Syllabus

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:

Lecturer


Matthias Kurzke
Email pmzmk@exmail.nottingham.ac.uk
Phone 0115 9514984


Bibliography


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


Note:

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Assessment



No assessment information is available yet.

No assignments have been set for this course.

Files


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Week(s)File
0CALCVAR.pdfL


Recorded Lectures


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