21.2 Antonia Diana (Naples): Long-time behavior and stability of surface diffusion flow

Abstract: In mathematics a geometric flow is a motion in time of some geometric object or structure, usually driven by a system of partial differential equations. Such geometric evolution equations have been applied to a variety of topological, analytical and physical problems, giving in some cases very fruitful results. We are actually interested in studying the surface diffusion flow of smooth hypersurfaces in the flat torus. According to this flow, smooth hypersurfaces move with the outer normal velocity given by the Laplacian of their mean curvature. A first local-in-time existence (and uniqueness) theorem was shown by Escher et al., then a long-time existence result, in dimension three, was presented in a paper by Acerbi et al. Even if the three-dimensional case is the most relevant from the physical point of view, since it describe the evolution in time of interfaces between solid phases of a system, driven by the surface diffusion of atoms under the action of a chemical potential, we mainly focus our attention on the generalization of these results to arbitrary dimension.

In our recent work (with C. Mantegazza and N. Fusco), we show that if the initial set is sufficiently "close" to a strictly stable
critical set for the Area functional, under a volume constraint, then the flow actually exists for all times and asymptotically converges in a suitable sense to a "translated" of the critical set.