MATJ5108 MA2: Geometric Evolutions Problems (JSS30) (1 cr)

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Study level:

Postgraduate studies

Grading scale:

Pass - fail

Language:

English

Responsible organisation:

Faculty of Mathematics and Science

Curriculum periods:

2020-2021, 2021-2022

Description

Among geometric evolution problems the motion of a surface according to its mean curvature is the best known and it has been widely studied in the last four decades. Since singularities may appear during the evolution, several weak formulations have been proposed to describe the long time behavior of surfaces. One of the possibilities is represent the initial surface as the level set of an auxiliary (initial) function and then to let evolve all the level set of such a function according to the same geometric law. This procedure transforms the original geometric evolution problem into an initial value problem for a suitable degenerate parabolic PDE, which can be treated using the machinery of viscosity solutions. This is the so-called level set approach and the course will mainly focus on it (and on the special case of the mean curvature flow), by introducing also the main notions and results concerning viscosity solutions needed for the theory.

As time permits, in the last part of the course we will also present more recent developments. Topics may include: phase-field approximations, minimizing movements and generalized (possibly nonlocal) curvature motions.

Learning outcomes

Students understand the concept of mean curvature flow and the basic geometric concepts related to it. Student is aware of the problems, which are related to the mean curvature flow such as appearance of singularities and the uniqueness of the solution. They know the definition of the viscosity solution and its basic properties such as comparison principle and are able to apply them.

Description of prerequisites

The course requires the knowledge of mathematical analysis in master student level. In particular, basics in partial differential equations. Basics in geometry is recommended.

Literature

Y. Chen, Y. Giga, and S. Goto, "Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations", J. Differential Geom. 33 (1991), 749–786.
L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), 635–681.
A. Chambolle, M. Morini, and M. Ponsiglione, "Nonlocal curvature flows", Arch. Ration. Mech. Anal. 218 (2015), 1263–1329

Completion methods

Method 1

Evaluation criteria:

Examination at the end of the course.

Select all marked parts

Parts of the completion methods

Participation in teaching (1 cr)

Type:

Participation in teaching