The convergence of solutions of anisotropic Allen–Cahn equations is studied when the interface thickness parameter (denoted by ε) tends to zero. It is shown that the convergence to a level set solution of the corresponding anisotropic interface equations is uniform with respect to the derivatives of a surface energy density function. As an application the crystalline motion of interfaces is shown to be approximated by the anisotropic Allen–Cahn equations.
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Yoshikazu Giga, Reiner Schätzle, Takeshi Ohtsuka, On a uniform approximation of motion by anisotropic curvature by the Allen–Cahn equations. Interfaces Free Bound. 8 (2006), no. 3, pp. 317–348DOI 10.4171/IFB/146