Calculus of Variations and Applications

Dr. Georgios Psaradakis (B6 - Office C4.03)


Mo. 15:30 - 17:00 in B6, 26 - A305

Do. 13:45 - 15:15 in B6, 26 - A304


Do. 15:30 - 17:00 in B6, 26 - A305

Content: In this course we focus on minimization problems that involve integral functionals which are defined on scalar functions, which are further defined on an open bounded domain Ω of the n-dimensional Euclidean space ℜ^n. Thus, our goal is to minimize F[u], where

F[u]:=∫_Ωf(x,u(x),∇u(x))dx, u:Ω→ℜ,

under certain conditions on the boundary values of u, on f:Ω×ℜ×ℜ^n→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 f=f(x,u,ξ), conditions on existence and uniqueness are presented. For instructive reasons, the difficult question of regularity of minimizers is detailed only for f=|ξ|^p/p, p>1.

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)

Course calendar (updated on 14/11/2019)

Lebesgue-measure/measurable functions/integral

A simple proof of the Poincare' inequality