Variational Methods in Analysis

Lecturers: Andreas Deuchert and Alessandro Olgiati

Lecture: Tuesday and Thursday from 3:00 - 4:45 pm
Exercises class: Monday from 3:00 - 4:45 pm

ECTS: 9 points

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.

Prior Knowledge: Analysis 1-3, Linear Algebra 1

Course Material: Handwritten lecture notes

Learning outcome: Understanding of advanced techniques in analysis that are necessary for the understanding of modern pde theory, mathematical physics, and geometric analysis.

Lecture notes:

  1. Part 1, Chapter 0 and 1
  2. Part 1, Chapter 2
  3. Part 1, Chapter 3
  4. Part 1, Chapter 4
  5. Part 1, Chapter 5
  6. Part 2