Sharp interface models for grain boundaries: derivation and approximation

Locally periodic structures show regions (grains) with different orientations and at the boundaries between grains there is the appearance of defects. This happens in physical systems (for instance at microscopic scales for metals or patterns in block copolymers) as well as in more geometric models (as local tassellations for partions and clusters, or optimal location problems). In all these cases the energy governing the systems concentrates at the grain boundaries. The understanding of this “surface tesion” is a key ingredient in order to reduce the complexity of the problem and work in a so to say sharp interface model.

I will present on one hand a derivation of this model in two dimension which account for the elastic distortion due to the presence of crystal defects (dislocations) at the interface, in the spirit of the model proposed by Lauteri and Luckhaus. On the other hand I will give an approximation result based on a phase field regularization of the sharp interface which is motivated by image reconstruction of polycrystals.