Functional Analysis

Geometric Measure Theory in Applications

Winter 2012/2013
Graduate Seminar on Analysis S4B1
M. Helmers

Seminar sessions Tuesday 14-16 in room 2.040 (Endenicher Allee 60).

Synopsis

Many tools in geometric measure theory (GMT) have been developed to generalise the notion of a smooth surface in order to deal with Plateau's problem of finding surfaces of minimal area. More generally, GMT studies geometric properties of sets, measures, and generalised surfaces; its methods have proved useful in many areas of mathematics including Geometry, Calculus of Variations, PDEs, and their applications. An overview of the subject can be found in the Encyclopedia of Mathematics [1].

In the seminar, we will study rectifiable sets, currents, and varifolds. The focus will be on varifolds and their application to problems involving curvature such as the Willmore energy, mean curvature flow, models for phase transitions, or the Helfrich bending energy of biological membranes.

Prerequisites

Basic knowledge of measure theory (measures, integration, convergence theorems) is essential, knowledge of submanifolds in the Euclidean space is nice but not necessary.

Organisation

Interested students please contact me by email. A preliminary meeting to finalise arrangements will be held end of September or early in October.

Literature

1. Geometric measure theory. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Geometric_measure_theory
2. W. K. Allard. On the first variation of a varifold. Annals of Mathematics 95, 417–491, 1972.
3. K. Brakke. The motion of a surface by its mean curvature. Mathematical Notes, 20, Princeton University Press, 1978.
4.  G. Bellettini and L. Mugnai. Approximation of Helfrich's functional via diffuse interfaces. SIAM J. Math. Anal. 42(6), 2402–2433, 2010.
5. J. E. Hutchinson. Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math J. 35(1), 45–71, 1986.
6. T. Ilmanen. Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature. J. Differential Geom. 38(2), 417–461, 1993.
7.  P. Mattila. Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability. Cambridge studies in advanced mathematics, vol. 44, 1995.
8. M. Röger and Y. Tonegawa. Convergence of phase-field approximations to the Gibbs-Thomson law. Calc. Var. Partial Differential Equations 32(1), 111–136, 2008.
9. L. Simon. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1(2), 281–326, 1993.
10. L. Simon. Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, 1983.
11. Y. Tonegawa. Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh 132A, 993–1019, 2002.