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.
Keywords: Anisotropic mean curvature flow, self-similar solutions, Long time behavior
@article{COCV_2021__27_1_A99_0,
author = {Cesaroni, A. and Kr\"oner, H. and Novaga, M.},
title = {Anisotropic mean curvature flow of {Lipschitz} graphs and convergence to self-similar solutions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021096},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021096/}
}
TY - JOUR AU - Cesaroni, A. AU - Kröner, H. AU - Novaga, M. TI - Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021096/ DO - 10.1051/cocv/2021096 LA - en ID - COCV_2021__27_1_A99_0 ER -
%0 Journal Article %A Cesaroni, A. %A Kröner, H. %A Novaga, M. %T Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021096/ %R 10.1051/cocv/2021096 %G en %F COCV_2021__27_1_A99_0
Cesaroni, A.; Kröner, H.; Novaga, M. Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 97. doi: 10.1051/cocv/2021096
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Cité par Sources :
The authors were supported by the INDAM-GNAMPA and by the PRIN Project 2019/24 Variational methods for stationary and evolution problems with singularities and interfaces.
