Geometric Measure Theory in Applications
Graduate Seminar on Analysis S4B1
Seminar sessions Tuesday 14-16 in room 2.040 (Endenicher Allee 60).
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 .
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.
Basic knowledge of measure theory (measures, integration, convergence theorems) is essential, knowledge of submanifolds in the Euclidean space is nice but not necessary.
Interested students please contact me by email. A preliminary meeting to finalise arrangements will be held end of September or early in October.
- Geometric measure theory. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Geometric_measure_theory
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- K. Brakke. The motion of a surface by its mean curvature. Mathematical Notes, 20, Princeton University Press, 1978.
- G. Bellettini and L. Mugnai. Approximation of Helfrich's functional via diffuse interfaces. SIAM J. Math. Anal. 42(6), 2402–2433, 2010.
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- T. Ilmanen. Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature. J. Differential Geom. 38(2), 417–461, 1993.
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- 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.
- L. Simon. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1(2), 281–326, 1993.
- L. Simon. Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, 1983.
- Y. Tonegawa. Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh 132A, 993–1019, 2002.