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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 2016 (Monday, October 3 to Friday, December 9)

Timetable

  • Fri 13:05 - 13:55

Prerequisites

Undergraduate calculus

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

Lecturers


Matthias Kurzke (main contact)
Email pmzmk@exmail.nottingham.ac.uk
Phone 0115 9514984
vcard
Yves van Gennip
Email Y.Vangennip@nottingham.ac.uk
Phone 0115 8466166
vcard


Students


Photo of Hussien Abugirda
Hussien Abugirda
(Reading)
Photo of Hayder Al-Saedi
Hayder Al-Saedi
(Loughborough)
Photo of Manal Alqhtani
Manal Alqhtani
(Birmingham)
Photo of Birzhan Ayanbayev
Birzhan Ayanbayev
(Reading)
Photo of Marco Baffetti
Marco Baffetti
(Nottingham)
Photo of Panagiota Birmpa
Panagiota Birmpa
(Sussex)
Photo of Blesson Chacko
Blesson Chacko
(Loughborough)
Photo of Alex Close
Alex Close
(Surrey)
Photo of Liam Dobson
Liam Dobson
(Newcastle)
Photo of Joe Driscoll
Joe Driscoll
(Leeds)
Photo of Duc Lam Duong
Duc Lam Duong
(Sussex)
Photo of Megan Goode
Megan Goode
(*External)
Photo of Alex Hiles
Alex Hiles
(Manchester)
Photo of Abdul Jumaat
Abdul Jumaat
(Liverpool)
Photo of Blaine Keetch
Blaine Keetch
(Nottingham)
Photo of James Kulmer
James Kulmer
(York)
Photo of Amy Mallinson
Amy Mallinson
(Manchester)
Photo of George Morrison
George Morrison
(Sussex)
Photo of Entesar Nasr
Entesar Nasr
(Liverpool)
Photo of Tom O'Neill
Tom O'Neill
(Surrey)
Photo of George Papanikos
George Papanikos
(Nottingham)
Photo of Sam Povall
Sam Povall
(Liverpool)
Photo of Michael Roberts
Michael Roberts
(Liverpool)
Photo of David Robertson
David Robertson
(Newcastle)
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Aleksandra Ross
(Sussex)
Photo of Anhad Sandhu
Anhad Sandhu
(Exeter)
Photo of Tomas Stary
Tomas Stary
(Exeter)
Photo of Ville Syrjanen
Ville Syrjanen
(Sussex)
Photo of Anthony Thompson
Anthony Thompson
(Liverpool)
Photo of Andrew Turner
Andrew Turner
(Birmingham)
Photo of Luke Warren
Luke Warren
(Nottingham)
Photo of Paul Wileman
Paul Wileman
(Nottingham)
Photo of Anthony Williams
Anthony Williams
(Manchester)
Photo of Jingsi Xu
Jingsi Xu
(Manchester)
Photo of Daoping Zhang
Daoping Zhang
(Liverpool)


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


Note:

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Assessment



This course will be assessed by a single take home paper. The paper will become available in January and answers will have to be submitted online within 2 weeks. The paper will be organised in questions, and marks will be given for answers. Half of the marks will be needed to pass.

Calculus of Variations take-home exam

Files:Exam paper
Released: Monday 9 January 2017 (44.6 days ago)
Deadline: Sunday 22 January 2017 (30.6 days ago)


Files


Files marked L are intended to be displayed on the main screen during lectures.

Week(s)File
0Complete Lecture notes (1-10)L
0Handwritten Remarks L1
0Handwritten Remarks L2
0Handwritten Remarks L4
0Lecture 1L
0Lecture 2L
0Lecture 3L
0Lecture 4L
0Lecture 5L
0Lecture 6L
0Lecture 7L


Recorded Lectures


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