For a nonsmooth positively one-homogeneous convex function phi : R-n --> [0, +infinity], it is possible to introduce the class R-phi(R-n) of smooth boundaries with respect to Q, to define their phi -mean curvature kappa (phi), and to prove that, for E epsilon R phi (R-n), kappa (phi) epsilon L-infinity(partial derivative E) [9]. Based on these results, we continue the analysis on the structure of partial derivative E and on the regularity properties of kappa (phi). We prove that a facet F of partial derivative E is Lipschitz (up to negligible sets) and that Kg has bounded variation on F. Further properties of the jump set of Kd are inspected: in particular, in three space dimensions, we relate the sublevel sets of kappa (phi) on F to the geometry of the Wulff shape W-phi := {phi less than or equal to 1}.
On a crystalline variational problem, Part II: BV regularity and structure of minimizers on facets
NOVAGA, MATTEO;
2001
Abstract
For a nonsmooth positively one-homogeneous convex function phi : R-n --> [0, +infinity], it is possible to introduce the class R-phi(R-n) of smooth boundaries with respect to Q, to define their phi -mean curvature kappa (phi), and to prove that, for E epsilon R phi (R-n), kappa (phi) epsilon L-infinity(partial derivative E) [9]. Based on these results, we continue the analysis on the structure of partial derivative E and on the regularity properties of kappa (phi). We prove that a facet F of partial derivative E is Lipschitz (up to negligible sets) and that Kg has bounded variation on F. Further properties of the jump set of Kd are inspected: in particular, in three space dimensions, we relate the sublevel sets of kappa (phi) on F to the geometry of the Wulff shape W-phi := {phi less than or equal to 1}.
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