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

Suppose that ${\Gamma }_{0}\subset {ℝ}^{n+1}$ is a closed countably $n$-rectifiable set whose complement ${ℝ}^{n+1}\setminus {\Gamma }_{0}$ consists of more than one connected component. Assume that the $n$-dimensional Hausdorff measure of ${\Gamma }_{0}$ 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 ${\Gamma }_{0}$. 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 ${\Gamma }_{0}\subset {ℝ}^{n+1}$ est un ensemble dénombrable fermé $n$-rectifiable dont le complément ${ℝ}^{n+1}\setminus {\Gamma }_{0}$ n’est pas connexe. Nous assumons que la mesure de Hausdorff $n$-dimensionnelle de ${\Gamma }_{0}$ 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 ${\Gamma }_{0}$. 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.

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
@article{AIF_2017__67_1_43_0,
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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