MATJ5108 MA2: Geometric Evolutions Problems (JSS30) (1 cr)
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.
Description of prerequisites
- 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