Year: 2025
Type: article
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 $$n=3$$ n = 3 by Acerbi, Fusco, Julin and Morini in [1] (extending previous results for spheres of Escher, Mayer and Simonett [15], Wheeler [37, 38] and Elliott and Garcke [14]). Our work generalizes such conclusion to any dimension $$n\in \mathbb {N}$$ n ∈ N . For sake of clarity, we show all the details in dimension $$n=4$$ 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.
Authors: Antonia Diana, Nicola Fusco, Carlo Mantegazza
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4416975517