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Calculus of Variations and Applications

General information

Lecturer: Dr. Georgios Psaradakis (B6 - Office C4.03)
Lectures (updated): Mo. 15:30 - 17:00 in B6, 26 - A305, Do. 13:45 - 15:15 in B6, 26 - A304
Tutorial: Do. 15:30 - 17:00 in B6, 26 - A305
Language: English
Prerequisites: Analysis I,II.

It is a master's course, however, all bachelor students who have already „Analysis I and II“ are welcome to join.

The main references are

  • Dacorogna, B. Introduction to the Calculus of Variations. 3rd ed. Imperial College Press 2015,

and chapter 5 and 8 from

  • Evans, L. C. Partial Differential Equations. 2nd ed. American Mathematical Society 2010

A notable new book in the field is

  • Rindler, F. Calculus of Variations. Springer 2018

You can get this last one in the following SpringerLink (internet connection provided by the Univ. Mannheim is required)

https://link.springer.com/content/pdf/10.1007%2F978-3-319-77637-8.pdf

Contents

In this course we focus on minimization problems that involve integral functionals which are defined on scalar functions, which are in turn defined on a sufficiently smooth, open and bounded domain Ω of the n-dimensional Euclidean space ℜ^n. Thus, our goal is to minimize I[u], where

I[u]:=∫_ΩL(∇u(x),u(x),x)dx, u:Ω→ℜ,

under certain conditions on the boundary values of u, on f:ℜ^n×ℜ×Ω→ℜ and possibly under further constraints. The basic questions on these problems are existence, uniqueness and regularity of minimizers, and the aim of the course is to be exposed to the basic theory underlying these questions. Under precise assumptions on the function L=f(ξ,u,x), conditions on existence and uniqueness are presented. For instructive reasons, the difficult question of regularity of minimizers is detailed only for L=|ξ|^p/p, p>1.

Lebesgue-measure/measurable functions/integral

A simple proof of the Poincare' inequality