no. 1
On the mean curvature flow of grain boundaries
[Sur le flot de la courbure moyenne des joints de grains]
Kim, Lami1 ; Tonegawa, Yoshihiro1
1 Department of Mathematics Tokyo Institute of Technology Ookayama, 152-8550 (Japan)
Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 43-142.

Supposons que Γ 0 ⊂ℝ n+1 est un ensemble dénombrable fermé n-rectifiable dont le complément ℝ n+1 ∖Γ 0 n’est pas connexe. Nous assumons que la mesure de Hausdorff n-dimensionnelle de Γ 0 est finie ou sa croissance est au plus exponentielle. Nous prouvons l’existence globale du flot de la courbure moyenne au sens de Brakke au départ de Γ 0 . Il existe une famille finie d’ensembles ouverts qui se déplacent d’une manière continue par rapport à la mesure de Lebesgue et dont les bords coïncident avec le support du flot de la courbure moyenne.

Suppose that Γ 0 ⊂ℝ n+1 is a closed countably n-rectifiable set whose complement ℝ n+1 ∖Γ 0 consists of more than one connected component. Assume that the n-dimensional Hausdorff measure of Γ 0 is finite or grows at most exponentially near infinity. Under these assumptions, we prove a global-in-time existence of mean curvature flow in the sense of Brakke starting from Γ 0 . There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time support of the mean curvature flow.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3077
Classification : 53C44, 49Q20
Keywords: mean curvature flow, varifold, geometric measure theory
Mots-clés : mouvement par courbure moyenne, varifold, théorie de la mesure géométrique

Kim, Lami 1 ; Tonegawa, Yoshihiro 1

1 Department of Mathematics Tokyo Institute of Technology Ookayama, 152-8550 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Kim, Lami; Tonegawa, Yoshihiro. On the mean curvature flow of grain boundaries. Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 43-142. doi : 10.5802/aif.3077. https://aif.centre-mersenne.org/articles/10.5802/aif.3077/

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