Calculus of Variations and Geometric Measure Theory

A. Diana - N. Fusco - C. Mantegazza

Stability for the Surface Diffusion Flow

created by root on 08 Apr 2023
modified by diana1 on 23 Oct 2023


Submitted Paper

Inserted: 8 apr 2023
Last Updated: 23 oct 2023

Year: 2023


We study the global existence and stability of surface diffusion flow (the normal velocity is given by the Laplacian of the mean curvature) of smooth boundaries of subsets of the $n$--dimensional flat torus. More precisely, we show that if a smooth set is ``close enough'' to a strictly stable critical set for the Area functional under a volume constraint, then the surface diffusion flow of its boundary hypersurface exists for all time and asymptotically converges to the boundary of a ``translated'' of the critical set. This result was obtained in dimension $n=3$ by Acerbi, Fusco, Julin and Morini (extending previous results for spheres of Escher, Mayer and Simonett, Wheeler, Elliott and Garcke). Our work generalizes such conclusion to any dimension $n\in\mathbb{N}$. For sake of clarity, we show all the details in dimension $n=4$ and we list the necessary modifications to the quantities involved in the proof in the general $n$--dimensional case, in the last section.