Geometric Variational Problems

The Plateau problem has been one of the most influential geometric variational problems; for instance, this concerns the areas geometric analysis, geometric measure theory, elliptic partial differential equations, and differential geometry. It consists of finding the surface of least m dimensional area amongst all surfaces spanning a given boundary in Euclidean space.

The lecture of consists of four parts: multilinear algebra (following [1]), basic geometric measure theory (Hausdorff densities, Hausdorff distance, Kirszbraun's extension theorem, tangent spaces and relative differentials for closes sets, area formula, rectifiable sets; following [1], [2], [3]), varifolds (rectifiable varifolds, first variation, radial deformations, and rectifiability theorem; following [4]), and a modern presentation of Reifenberg's approach to the Plateau problem (simplifying current research papers).  In regard to the latter, our axiomatic treatment will allow the present lecture to focus on the analysis part of the problem.

1. Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. https://dx.doi.org/10.1007/978-3-642-62010-2
2. Ulrich Menne. Real analysis. Lectures notes, University of Potsdam, 2015.
3. Ulrich Menne. Einführung in die Geometrische Maßtheorie. Lectures notes, University of Potsdam, 2015.
4. William K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417--491, 1972. https://dx.doi.org/10.2307/1970868


Date and time info
Monday & Friday, 9:15 - 10:45

Prerequisites
Real analysis (in particular, Hausdorff measure & Rademacher's theorem)

Audience
Diploma & PhD students

Language
English

Remarks and notes
Lectures notes for the last part shall be created as the course progresses.