Asymptotic solutions for the equilibrium crystal shape with small corner energy regularization

Dynamic models for facet formation often employ a regularization of the surface energy based on a corner energy term. Here we consider the effect of this regularization on the equilibrium shape of a solid particle in two dimensions. Using matched asymptotic expansions we determine the explicit solution for the corner shape in the presence of the regularization. Our results show that for a class of surface energy anisotropy models the regularized solution approaches the classic sharp-corner results as the regularization approaches zero. The results validate the use of the regularization in numerical calculations for the equilibrium problem. Finally, a byproduct of the analysis is an exact solution for the equilibrium shape of a semi-infinite wedge in the presence of the regularization.