Calculus of Variations

Webpage of the Course MAT.494UB SS 2020/21

General Information

Welcome to the Course of Calculus of Variations MAT.494UB for the Master Degree in Mathematics at the University of Graz

• Lecturer: Dr Silvio Fanzon
• Email: Silvio.Fanzon@uni-graz.at
• Office: Room 501, Department of Mathematics & Scientific Computing
• Office hours: Flexibly arranged online, due to COVID-19

Please do not hesitate to email me with any questions you have regarding the module or the exercises. I will answer your questions at the beginning of each class, or schedule online office hours

Lectures Calendar

Lectures will be weekly and last approximately 90 minutes. There will be 15 classes in total, with dates

• 3, 10, 17, 24 March 2021
• 14, 21, 28 April 2021
• 5, 12, 19, 26 May 2021
• 2, 9, 16, 23 June 2021

Lectures will be online, due to the current COVID-19 Orange Light at the University of Graz. Recordings will be available on this webpage as well as on the Moodle page of the course. Lecture notes and syllabus are released weekly on this page

Topics

• Essential Topics: Minimization of integral functionals for maps of one variable. First variation and indirect method. Metric spaces. Gateaux and Frechet derivatives in normed spaces. Fundamental Lemma of Calculus of Variations. Du Boys-Reymond Lemma. Euler-Lagrange equations for unconstrained problems. Boundary conditions. Direct method and Hilbert spaces. Strong and weak convergence. Weak compactness of unit ball. One dimensional Sobolev spaces. Direct method in Sobolev spaces and applications. Relaxation in metric spaces. Convex envelopes. Gamma convergence with model applications.

• Optional Topics: calibrations, Weierstrass fields, functions of bounded variation in one variable, $\Gamma$-convergence

• Previous knowledge: Basic knowledge of real analysis, \(L^p\) spaces, general topology and functional analysis.

References

• Calculus of Variations: I will follow , , and the wonderful course by Gobbino
• Practical Calculus of Variations: A good reference with lots of examples and guided exercises is Kielhoefer
• Functional Analysis: All the topics discussed throughout the course can be found in
• Calculus in Normed Spaces: Mainly following . I had to take a couple of lemmas from (in French), but those are quite quick
• Convex Analysis: I refer to and
• \(L^p\) spaces: I will follow the construction of
• Sobolev spaces: The main reference is . I will also take something from
• \(\Gamma\)-convergence: The book by Braides

Assessment

There will be an oral examination of about 1 hour on the topics of the course. This will happen online, on a mutually agreed day in June or July.

Lecture Notes and Videos

For a detailed account of the topics of each lesson, please refer to the syllabus

• Syllabus: Download in PDF

Lecture Notes and Video Recordings of lectures are available at the links below

Youtube playlist with all the video recordings available here

Date	Lecture Notes	Video Recordings	Topics
3 March	Lesson 1	Part 1     Part 2	Introduction. Basic examples. Functional analysis revision
10 March	Lesson 2	Part 1     Part 2	Functional Analysis Revision. Calculus in Normed Spaces
17 March	Lesson 3	Part 1     Part 2	Calculus in Normed Spaces. Indirect Method
24 March	Lesson 4	Part 1     Part 2	Fundamental Lemmas. Boundary conditions
14 April	Lesson 5	Part 1     Part 2	Euler-Lagrange Equation
Extra	Revision	Video overview	Revision of \(L^p\) spaces
21 April	Lesson 6	Part 1     Part 2	Sufficient Conditions: convexity, trivial lemma. Convolutions
28 April	Lesson 7	Part 1     Part 2	FLCV and DBR Lemma. Sobolev spaces
5 May	Lesson 8	Part 1     Part 2	Sobolev Spaces: regularity and density results
12 May	Lesson 9	Part 1     Part 2	Sobolev embedding. Ascoli-Arzelà Theorem
19 May	Lesson 10	Part 1     Part 2	Higher order Sobolev Spaces. Traces. Euler-Lagrange Equation
26 May	Lesson 11	Part 1     Part 2	Boundary conditions. Sufficient conditions. Direct Method
2 June	Lesson 12	Part 1     Part 2	Direct method: example. General existence theorem
9 June	Lesson 13	Part 1     Part 2	LSC Envelope. Relaxation and its computation
16 June	Lesson 14	Part 1     Part 2	Relaxation of integral functionals. \(\Gamma\)-convergence
23 June	Lesson 15	Part 1     Part 2	Examples of \(\Gamma\)-convergence. Homogenization problems

Exercise Sheets

This course has a companion practical course MAT.495UB taught by Dr Cinzia Soresina. The exercises assigned will complement the theory seen in the main course, and are released every two weeks. Although I have authored most of the exercises, these will not be assessed in my course. These are the instructions by Dr Soresina.

Due date	Exercise Sheet	Topics
12 March	Sheet 1	Metric spaces
26 March	Sheet 2	Normed spaces. Compactness. Weak topologies
23 April	Sheet 3	Frechet derivative. Gateaux derivative
7 May	Sheet 4	Convolutions. Cut-off. Smooth approximations. Minimization
28 May	Sheet 5	Sobolev spaces. Poincaré Inequality
11 June	Sheet 6	Assumptions of Theorem 9.9. Brachistochrone
25 June	Sheet 7	Relaxation. \(\Gamma\)-convergence