Nonlocal diffusion of smooth sets

Anoumou Attiogbe; Mouhamed Moustapha Fall; El Hadji Abdoulaye Thiam; Anoumou Attiogbe; Mouhamed Moustapha Fall; El Hadji Abdoulaye Thiam

doi:10.3934/mine.2022009

Mathematics in Engineering

2022, Volume 4, Issue 2: 1-22. doi: 10.3934/mine.2022009

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Nonlocal diffusion of smooth sets

1.
African Institute for Mathematical Sciences in Senegal, KM 2, Route de Joal, B.P. 14 18. Mbour, Senegal
2.
Université Iba Der Thiam de Thies, UFR des Sciences et Techniques, département de mathématiques, Thies

Received: 13 January 2021 Accepted: 10 May 2021 Published: 11 June 2021

We consider normal velocity of smooth sets evolving by the $ s- $fractional diffusion. We prove that for small time, the normal velocity of such sets is nearly proportional to the mean curvature of the boundary of the initial set for $ s\in [\frac{1}{2}, 1) $ while, for $ s\in (0, \frac{1}{2}) $, it is nearly proportional to the fractional mean curvature of the initial set. Our results show that the motion by (fractional) mean curvature flow can be approximated by fractional heat diffusion and by a diffusion by means of harmonic extension of smooth sets.
- motion by fractional mean curvature flow,
- fractional heat equation,
- fractional mean curvature,
- harmonic extension
Citation: Anoumou Attiogbe, Mouhamed Moustapha Fall, El Hadji Abdoulaye Thiam. Nonlocal diffusion of smooth sets[J]. Mathematics in Engineering, 2022, 4(2): 1-22. doi: 10.3934/mine.2022009

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Abstract

We consider normal velocity of smooth sets evolving by the $ s- $fractional diffusion. We prove that for small time, the normal velocity of such sets is nearly proportional to the mean curvature of the boundary of the initial set for $ s\in [\frac{1}{2}, 1) $ while, for $ s\in (0, \frac{1}{2}) $, it is nearly proportional to the fractional mean curvature of the initial set. Our results show that the motion by (fractional) mean curvature flow can be approximated by fractional heat diffusion and by a diffusion by means of harmonic extension of smooth sets.

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