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.

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

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




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

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

The lecture notes are available for download below.

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