Existence of solutions to an anisotropic phase-field model

We consider an anisotropic phase-field model for the isothermal solidification of a binary alloy due to Warren-Boettinger ( Acta. Metall. Mater. 1995; 43(2):689). Existence of weak solutions is established under a certain convexity condition on the strongly non-linear second-order anisotropic operator and Lipschitz and boundedness assumptions for the non-linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non-linearities. The qualitative properties of the solutions are illustrated by a numerical example