On a crystalline variational problem, Part II : BV regularity and structure of minimizers on facets

• Published in: Archive for rational mechanics and analysis 2001-04, Vol.157 (3), p.193-217
• Main Authors: BELLETTINI, G, NOVAGA, M, PAOLINI, M
• Format: Article
• Language: English
• Subjects: Cross-disciplinary physics: materials science; rheology; Exact sciences and technology; Materials science; Methods of crystal growth; physics of crystal growth; Phase diagrams and microstructures developed by solidification and solid-solid phase transformations; Physics; Theory and models of crystal growth; physics of crystal growth, crystal morphology and orientation
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s002050100126

For a nonsmooth positively one-homogeneous convex function Φ:^sup n^[arrow right] [0,+∞[, it is possible to introduce the class ?^sub Φ^ (^sup n^) of smooth boundaries with respect to Φ, to define their Φ-mean curvature κ^sub Φ^, and to prove that, for E?^sub Φ^ (^sup n^), κ^sub Φ^L ^sup ∞^(δE) [9]. Based on these results, we continue the analysis on the structure of δE and on the regularity properties of κ^sub Φ^. We prove that a facet F of δE is Lipschitz (up to negligible sets) and that κ^sub Φ^ has bounded variation on F. Further properties of the jump set of κ^sub Φ^ are inspected: in particular, in three space dimensions, we relate the sublevel sets of κ^sub Φ^ on F to the geometry of the Wulff shape ?^sub Φ^{Φ≤ 1 }.[PUBLICATION ABSTRACT]