A Crystalline Motion: Uniqueness and Geometric Properties
Piotr Rybka · SIAM Journal on Applied Mathematics · 1997
The author studies a model of crystalline motion in the plane. Existence and uniqueness of local in time solutions are shown. Geometric properties of the flow are studied, assuming that the Wulff shape is a regular N-sided polygon. The author shows that a small, convex grain of ice immersed in the cold melt shrinks, provided that the flow does not overly deform the initial interface. The author is able to verify the last condition only if the initial interface is a scaled Wulff shape. In this case, the free boundary will remain a scaled Wulff shape at later times. This is shown by using an isoperimetric inequality.