# Calculus of Variations I and II

## Calculus of Variations II

The course Calculus of Variations I is designed to introduce students to a classical branch of mathematics. For this purpose, terms such as minimum, maximum and critical point, known from basic analysis lectures, will be extended and classic one-dimensional minimization problems will be presented. The students will learn how to formulate and solve minimization problems on their own.

Topics

Among others, the following topics are covered: Euler-Lagrange equations of one-dimensional variational integrals, Sobolev functions on bounded domains, Dirichlet principle, Compactness criteria, Lower semicontinuity, Existence theory, Regularity of weak solutions, Variational applications

Prerequisites

Prerequisites for taking the course are the modules Analysis I-III.

Examination

The examination consist of a written or oral exam; duration and type of the exam are announced at the beginning of the semester. For admission to the exam, a sufficient number of exercises has to be solved and submitted.

Literature

1. G. Buttazzo, M. Giaquinta, S. Hildebrandt: One Dimensional Variational Problems, Oxford University Press 1988;
2. U. Brechtken-Manderscheid: Einführung in die Variationsrechnung, Wissenschaftliche Buchgesellschaft 1983;
3. W. Rudin: Reelle und Komplexe Analysis, Oldenbourg Verlag 1999;

## Calculus of Variations II

Calculus of Variations II founds on Calculus of Variations I to introduce students to multi-dimensional calculus of variations. Several examples from physics and engineering sciences can be formulated as minimization problems. Basic methods for finding solutions of such problems will be imparted.

Topics

Among others, the following topics are covered: Euler-Lagrange equations, multi-dimensional variation integrals, Sobolev-functions on bounded domains, Dirichlet principle, criteria for compactness, lower semi-continuity, existence theorems, regularity of weak solutions.

Prerequisites

A prerequisite for taking the course is basic knowledge from Calculus of Variations I.

Examination

The examination consist of a written or oral exam; duration and type of the exam are announced at the beginning of the semester. For admission to the exam, a sufficient number of exercises has to be solved and submitted.

Literature

1. J. Jost, X. Li-Jost: Calculus of Variations, Cambridge University Press 1998
2. M. Giaquinta, S. Hildebrand: Calculus of Variations I, II, Springer-Verlag Berlin 1996
3. C.B. Morrey: Multiple Integrals in the Calculus of Variations, Springer-Verlag New York 1966