Some regularity results for minimal crystals

¹ Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy; ambrosio@emath.ethz.ch.
² Dipartimento di Matematica, Università di Pisa, via F. Buonarroti 2, 56126 Pisa, Italy.

Received: 4 October 2001

Abstract

We introduce an intrinsic notion of perimeter for subsets of a general Minkowski space (i.e. a finite dimensional Banach space in which the norm is not required to be even). We prove that this notion of perimeter is equivalent to the usual definition of surface energy for crystals and we study the regularity properties of the minimizers and the quasi-minimizers of perimeter. In the two-dimensional case we obtain optimal regularity results: apart from a singular set (which is -negligible and is empty when the unit ball is neither a triangle nor a quadrilateral), we find that quasi-minimizers can be locally parameterized by means of a bi-lipschitz curve, while sets with prescribed bounded curvature are, locally, lipschitz graphs.