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Variational Methods in Analysis (University of Zurich)
Last Updated: 2026-02-05 16:06:49
Objective
Understanding of variational methods that can be applied in many areas of mathematics as e.g. analysis of pde, mathematical physics and geometric analysis.
Content
General description In this lecture we are interested in problems where one needs to minimize or maximize a certain quantity (a functional) depending on a variable, which may be a collection of parameters, a function or a more general mathematical object (variational problems). Variational problems play an important role in several areas of modern mathematics as, e.g., analysis of pde, mathematical physics or geometric analysis. In many examples the functional depends on quantities that need to be varied in an infinite dimensional vector space (think of the Fourier series representation of a periodic function). Such expressions can show a rich phenomenology and advanced mathematical tools are needed to prove e.g. the existence/absence of a minimizer/maximizer and to make statements about their properties. The aim of this lecture is to familiarize the students with a mathematical toolbox that is appropriate for the study of such kind of problems. In the first part of the lecture we cover some advanced topics in analysis as e.g. Fourier transform, distributions (generalized functions), weak derivatives, Sobolev spaces, weak and strong convergence and Sobolev inequalities. Apart from their relevance for the study of variational problems, they are important tools in modern analysis and therefore also of independent interest. In the second part we introduce the audience to techniques from the calculus of variations. Although these techniques are very general, we will, for the sake of concreteness, introduce them in the framework of the Schrödinger equation and certain models originating from atomic physics. Topics to be covered are: Introduction to the direct method in the calculus of variations, weak lower semi-continuity, relaxation of variational problems and binding inequalities, methods based on convexity, uniqueness of minimizers, Euler—Lagrange equation, regularity of minimizers, spherically symmetric rearrangement, and non-convex problems.
Resources
Lecture Notes
Handwritten lecture notes
General Information
- Language
- English
- Levels
- BSC , MSC
Examination
- Type
- graded semester performance
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture |
Variational Methods in Analysis (University of Zurich)
**Course at University of Zurich**
|
No time listed | 4 h weekly |
| exercise |
Variational Methods in Analysis (University of Zurich)
**Course at University of Zurich**
|
No time listed | 2 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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