A characterization of convex calibrable sets in \(\mathbb R^N\) with respect to anisotropic norms. (English) Zbl 1144.52002
Summary: 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 “\(\varphi\)-calibrability” of bounded convex sets in \(\mathbb R^N\) with respect to a norm \(\varphi\) (called anisotropy in the sequel) by the anisotropic mean \(\varphi\)-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 \(\varphi\)-ball condition contains a convex \(\varphi\)-calibrable set \(K\) such that, for any \(V\in [|K|,|C|]\), the subset of \(C\) of volume \(V\) which minimizes the \(\varphi\)-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set.
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 35J70 | Degenerate elliptic equations |
| 49J40 | Variational inequalities |
| 35K65 | Degenerate parabolic equations |
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