Department Mathematik



Direct Methods in the Calculus of Variations

Lecturer: Dr. Soneji

Dates: Summer Semester 2014

Course Overview

The Calculus of Variations is a large and active field of modern mathematics. It has links to many other branches of mathematics, such as geometry and partial differential equations, and has applications in fields including physics, engineering and economics.

We shall begin this course by giving a "practical background'' in aspects of functional analysis, measure theory and Sobolev Spaces. The tools developed here are also an important grounding in many aspects of modern PDE theory.

Equipped with this machinery, we shall then consider variational integrals of the form

F(u;\Omega) = \int_{\Omega} f(\nabla u) dx .

We will investigate the problem of existence of minimisers of such integrals in the multi-dimensional, vectorial setting. This will lead us to consider notions of convexity and Morrey's theorem, which is the central result of this course.

Zeit und Ort: Do 10-12 HS B 039
Übungen: Fr 14-16 HS B 041
für: Mathematics masters students.
Vorkenntnisse: Analysis I-III
Functional Analysis helpful but not essential.
Leistungsnachweis: Gilt für Masterprüfung Mathematik (WP47.2+47.3), Diplomhauptprüfung Mathematik (AM).

Problem Sheets:

Past problem sheets may be requested by email.


B. Dacorogna, "Direct methods in the Calculus of Variations". Second edition (2008).
B. Dacorogna, "Introduction to the Calculus of Variations". Second edition (2009).
I. Fonseca and G. Leoni, "Modern Methods in the Calculus of Variations: Lp Spaces" (2007).