Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Towards a variational theory of phase transitions involving curvature

Roger Mosera1

a1 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK (r.moser@bath.ac.uk)

Abstract

An anisotropic area functional is often used as a model for the free energy of a crystal surface. For models of faceting, the anisotropy is typically such that the functional becomes non-convex, and then it may be appropriate to regularize it with an additional term involving curvature. When the weight of the curvature term tends to zero, this gives rise to a singular perturbation problem.

The structure of this problem is comparable to the theory of phase transitions studied first by Modica and Mortola. Their ideas are also useful in this context, but they have to be combined with adequate geometric tools. In particular, a variant of the theory of curvature varifolds, introduced by Hutchinson, is used in this paper. This allows an analysis of the asymptotic behaviour of the energy functionals.

(Received June 11 2010)

(Accepted June 22 2011)