Geometric Analysis - Einzelansicht

In this course we study the isoperimetric problem. This is the task to find a domain with prescribed volume and minimal perimeter. More precisely, the goal is to show that such a domain in Euclidean Space is the ball. We will cover several approaches to this problem in the classical setting depending on the class of domains considered. The methods range from Geometry to Analysis, and depend a lot on the regularity of the set of competing domains. In the class of domains with $$C^2$$-boundary, differential geometry helps, but with less assumed regularity (such as domains with corners), the analytical difficulties increase.

• Isaac Chavel, Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives, Cambridge University Press, 2001
• Lawrence C. Evans, Ronald F. Gariepy: Measure Theory and Fine Properties of Functions, CRC Press, 2015
• Antonio Ros: The Isoperimetric Problem, in Global theory of minimal surfaces, 2001

Essential: Measuretheory, Integration, Submanifolds.

Helpful: (Elementary) Differential geometry of curves and surfaces. If needed, this can be discussed in the course.

Presentation in the Seminar.