On the mean curvature flow of grain boundaries
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, p. 43-142

Suppose that Γ 0 n+1 is a closed countably n-rectifiable set whose complement 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.

Supposons que Γ 0 n+1 est un ensemble dénombrable fermé n-rectifiable dont le complément 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.

Received : 2015-11-22
Revised : 2016-04-04
Accepted : 2016-05-12
Published online : 2017-01-10
DOI : https://doi.org/10.5802/aif.3077
Classification:  53C44,  49Q20
Keywords: mean curvature flow, varifold, geometric measure theory
     author = {Kim, Lami and Tonegawa, Yoshihiro},
     title = {On the mean curvature flow of grain boundaries},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     pages = {43-142},
     doi = {10.5802/aif.3077},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_1_43_0}
Kim, Lami; Tonegawa, Yoshihiro. On the mean curvature flow of grain boundaries. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 43-142. doi : 10.5802/aif.3077. http://www.numdam.org/item/AIF_2017__67_1_43_0/

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