| Issue |
ESAIM: COCV
Volume 27, 2021
|
|
|---|---|---|
| Article Number | 97 | |
| Number of page(s) | 17 | |
| DOI | https://doi.org/10.1051/cocv/2021096 | |
| Published online | 12 October 2021 | |
Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions*
1
Department of Statistical Sciences, University of Padova Via Cesare Battisti 141,
35121
Padova, Italy.
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: annalisa.cesaroni@unipd.it
Received:
7
June
2021
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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