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.