Published Paper
Inserted: 8 apr 2023
Last Updated: 12 feb 2026
Journal: Calc. Var. PDE
Volume: 65
Number: 1
Pages: Paper No. 6, 34 pp.
Year: 2026
Abstract:
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.
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