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A.A. 2018-2019

Motion of Level Sets by Mean Curvature

V. Pagliari and M. Pozzetta

This is an informal lecture course based on the four papers by Evans and Spruck on this topic. Precisely:

• Motion of Level Sets by Mean Curvature I, J. Diff. Geom., 33, 1991, pp. 635-681.
• Motion of Level Sets by Mean Curvature II, Trans. American Math. Soc., 330(1), 1992, pp. 321-332.
• Motion of Level Sets by Mean Curvature III, J. Geom. Analysis, 2(2), 1992, pp. 121-150.
• Motion of Level Sets by Mean Curvature IV, J. Geom. Analysis, 5(1), 1995, pp. 77-114.

Lectures:

1. November 6th, 2018. Lecturer: Pagliari.

   In this first meeting, we outlined the overall content of the papers by Evans and Spruck. We recalled some basic notions as well, as the one of second fundandamental form and of mean curvature. Next, we discussed about the second paper by Evans and Spruck, which is concerned with a proof of existence for small times of classical solutions to the mean curvature motion. The strategy is based on the study of the parabolic PDE that is solved by the signed distance function from the evolving front, and the conclusion is achieved by a fixed-point argument.

   Further references:

   • L. Ambrosio: Geometric evolution problems, distance function and viscosity solutions, in Calculus of Variations and Partial Differential Equations: Topics on Geometrical Evolution Problems and Degree Theory, Springer Berlin Heidelberg, 2000, pp. 5-93.

2. November 12th, 2018. Lecturer: Pozzetta.

   In this lecture, we concluded the proof of the existence of a smooth solution to the PDE introduced in the first lecture. We also showed that the level sets of such a solution actually evolve smoothly according to the mean curvature flow, and, from this, we obtained some important classical theorems.

3. November 23th, 2018. Lecturer: Pagliari.

   We previously proved existence of classical solutions to the mean curvature motion, and we observed the onset of singularities in finite time. In this lecture we focused on a notion of viscosity solution suited for the problem at stake. In particular, we gave our definition in the framework of the level set formulation of the problem, and we discussed some basic properties of these class of weak solutions. We followed the first sections of the 1991 paper by Evans and Spruck.

4. November 26th, 2018. Lecturer: Pozzetta.

   We proved that the composition of a continuous function and of a weak solution is still a solution. Then, we introduced the concepts of inf/sup-convolutions, which can be thought as regularizations of a given function, and we proved their most useful properties in connection with weak solutions.

5. December 3rd, 2018. Lecturer: Pagliari.

   We proved a comparison principle for sub- and supersolutions to the level set equation of mean curvature motion. Namely, we showed that subsolutions stay below supersolutions, provided this holds at the initial instant t=0.

6. December 10th, 2018. Lecturer: Pozzetta.

   We proved existence of weak solutions of the mean curvature flow equation. The proof involves an approximation argument that exploits classical results about solutions of parabolic equations, which in turn appear as regularizations of the given flow. Once existence is on hand, we are in position to define the generalized mean curvature flow as the evolution of the zero level sets of the weak solution.

7. December 17th, 2018. Lecturer: Pagliari.

   We showed that the definition of generalized mean curvature flow is well posed, and that it is consistent with the classical motion if and so long as the latter exists.

8. Next lecture: January 9th, 2019. Lecturer: Pozzetta.

   We shall discuss some relevant qualitative properties of the generalized mean curvature flow, and then we shall study the evolution of initial data with positive mean curvature.