  A phase-field model for diffusion-limited crystal growth is formulated that
is capable of handling highly anisotropic interfaces. It uses a Willmore
regularization that yields corners of finite size. An asymptotic analysis
reveals that Herring's law is recovered for the advancing surfaces. The model
is validated by conducting simulations of dendritic growth for low anistorpies
and comparing the results to the data from the literature. The model makes it
possible to simulate high anisotropy dendrites for which the standard
phase-field models are ill-posed. In this regime, the interplay between a
Herring instability on the dendrite flanks and the corner regularization
creates zig-zag shaped corrugations and leads to a non-monotonic trend of tip
velocity as a function of anisotropy strength.
