A planar anisotropic curvature flow equation with constant driving force term is considered when the interfacial energy is crystalline. The driving force term is given so that a closed convex set grows if it is sufficiently large. If initial shape is convex, it is shown that a flat part called a facet (with admissible orientation) is instantaneously formed. Moreover, if the initial shape is convex and slightly bigger than the critical size, the shape becomes fully faceted in a finite time provided that the Frank diagram of interfacial energy density is a regular polygon centered at the origin. The proofs of these statements are based on approximation by crystalline algorithm whose foundation was established a decade ago. Our results indicate that the anisotropy of interfacial energy plays a key role when crystal is small in the theory of crystal growth. In particular, our theorems explain a reason why snow crystal forms a hexagonal prism when it is very small.
Mi-Ho Giga; Yoshikazu Giga. On the role of kinetic and interfacial anisotropy in the crystal growth theory. Interfaces and free boundaries, Tome 15 (2013) no. 4, pp. 429-450. doi: 10.4171/ifb/309
@article{10_4171_ifb_309,
author = {Mi-Ho Giga and Yoshikazu Giga},
title = {On the role of kinetic and interfacial anisotropy in the crystal growth theory},
journal = {Interfaces and free boundaries},
pages = {429--450},
year = {2013},
volume = {15},
number = {4},
doi = {10.4171/ifb/309},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/309/}
}
TY - JOUR
AU - Mi-Ho Giga
AU - Yoshikazu Giga
TI - On the role of kinetic and interfacial anisotropy in the crystal growth theory
JO - Interfaces and free boundaries
PY - 2013
SP - 429
EP - 450
VL - 15
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/309/
DO - 10.4171/ifb/309
ID - 10_4171_ifb_309
ER -
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%A Mi-Ho Giga
%A Yoshikazu Giga
%T On the role of kinetic and interfacial anisotropy in the crystal growth theory
%J Interfaces and free boundaries
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%P 429-450
%V 15
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/309/
%R 10.4171/ifb/309
%F 10_4171_ifb_309
