On a Crystalline Variational Problem, Part II:¶BV Regularity and Structure of Minimizers on Facets

Abstract:

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