Volume 27, 2021
|Number of page(s)||17|
|Published online||12 October 2021|
Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions*
Department of Statistical Sciences, University of Padova Via Cesare Battisti 141,
2 Universität Duisburg-Essen, Fakultät für Mathematik. Thea-Leymann-Straße9, 45127, Essen, Germany.
3 Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy.
** Corresponding author: firstname.lastname@example.org
Accepted: 26 September 2021
We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.
Mathematics Subject Classification: 53C44 / 35K93
Key words: Anisotropic mean curvature flow / self-similar solutions / long time behavior
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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