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

Pages: 697 - 707, Issue Special, September 2009

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Ken Shirakawa - Department of Applied Mathematics, Faculty of Engineering, Kobe University, 1-1 Rokkodai, Nada, Kobe, 657-8501, Japan (email)

Abstract: 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.

Keywords:  Allen-Cahn equation, anisotropy by crystalline type setting, stability analysis
Mathematics Subject Classification:  Primary: 35K90, 35B35; Secondary: 35J50

Received: July 2008;      Revised: March 2009;      Published: September 2009.