Calculus of Variations and Geometric Measure Theory

L. Ambrosio - M. Novaga - E. Paolini

Some Regularity Results for Minimal Crystals

created on 22 Feb 2000
modified by paolini on 22 Jul 2009


Published Paper

Inserted: 22 feb 2000
Last Updated: 22 jul 2009

Journal: ESAIM: COCV volume dedicated to J. L. Lions
Volume: 8
Number: 69--103
Year: 2002



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 mimimizers and the quasi-minimizers of perimeter. In the two-dimensional case we obtain optimal regularity results: apart from the singular set (which is $\mathcal{H}^1$-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.

Keywords: regularity, minimal surfaces, perimeter, crystals