Calculus of Variations and Geometric Measure Theory
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V. Caselles - A. Chambolle - S. Moll - M. Novaga

A characterization of convex calibrable sets in $\mathbb R^N$ with respect to anisotropic norms

created by novaga on 11 Nov 2018

[BibTeX]

Published Paper

Inserted: 11 nov 2018
Last Updated: 11 nov 2018

Journal: Ann. Inst. H. Poincaré - Anal. Non Linéaire
Volume: 25
Number: 4
Pages: 803-832
Year: 2008

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 $\phi$-calibrability of bounded convex sets in $\mathbb R^N$, with respect to a norm $\phi$ (called anisotropy), by the anisotropic mean $\phi$-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 $\phi$-ball condition contains a convex $\phi$-calibrable set $K$ such that, for any $V\in [
K
,
C
]$, the subset of $C$ of volume $V$ which minimizes the $\phi$-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data by the characteristic function of a bounded convex set.


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