# American Institute of Mathematical Sciences

2009, 2009(Special): 697-707. doi: 10.3934/proc.2009.2009.697

## Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies

 1 Department of Applied Mathematics, Faculty of Engineering, Kobe University, 1-1 Rokkodai, Nada, Kobe, 657-8501

Received  July 2008 Revised  March 2009 Published  September 2009

This paper is devoted to the stability analysis for two dimensional interfaces in solid-liquid phase transitions, represented by some types of Allen-Cahn equations. Each Allen-Cahn equation is derived from a free energy, associated with a two dimensional Finsler norm, under the so-called crystalline type setting, and then the Wulff shape of the Finsler norm is supposed to correspond to the basic structural unit of masses of pure phases (crystals). Consequently, special piecewise smooth Jordan curves, based on Wulff shapes, will be exemplified in the main theorems, as the geometric representations of the stability condition.
Citation: Ken Shirakawa. Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies. Conference Publications, 2009, 2009 (Special) : 697-707. doi: 10.3934/proc.2009.2009.697
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