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Admissible Crystal
•A simple Polygon•All outward normals belong to•The normal of adjacent facet is parallel to the normal adjacent in the Wulff Shape
N:
n1n2
n3
nm
ni
n1n2
n3
n2
Admissible
WulffShape
NotAdmissible n1
n3
Jump!
Crystalline Flow
Vi = g(ni;ÿiÉ (ni)=L i(t))
g(á;á) : Nondecreasing in 2nd variableÿi : Transition Number
-1 0 +1
É (ni) : Length of facet of Wulff Shape
É (n1)
L i(t)L i(t) : Length of i-th facet
É (n1)L i(t)
Nonlocal Curvature
Crystalline Flow
Facet Disappearing at t = T*
Case A:ÿj6=0•The polygon becomes convex near T* and all facet disappear at T*.
All facets disappear at t = T*.
Crystalline Flow
õ7! g(ni;õ)
limõ! æ1 g(ni;õ) =æ1 ni 2 N
g2 Dg
RIæg(ni;õ)õ
à2dõ =æ1
is locally Lipschitz on R nf 0g
õ7! g(ni;õ) is nondecreasing on R
for all
Case B does not occur if
I+ = (1;1 ); I à = (à 1 ;0)
Crystalline Flow
Facet Disappearing at t = T*
Case C: ÿj = 0•At most two consecutive facets disappear.
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