On an area-preserving crystalline motion

The asymptotic behavior of solutions to an area-preserving crystalline motion is investigated in this paper. In this equation, the area enclosed by the solution polygon is preserved, while its circumference keeps on shrinking. By geometric consideration, establishing several isoperimetric inequalities and using the theory of dynamical systems, we show that the asymptotic shape of a solution polygon is the boundary of the Wulff shape. Received: 23 February 2000 / Accepted: 23 January 2001 / Published online: 23 April 2001