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