Abstract
A set is called “calibrable” if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the “ϕ-calibrability” of bounded convex sets in RN with respect to a norm ϕ (called anisotropy in the sequel) by the anisotropic mean ϕ-curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body C satisfying a ϕ-ball condition contains a convex ϕ-calibrable set K such that, for any V∈[|K|,|C|], the subset of C of volume V which minimizes the ϕ-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set.
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