Calculus of Variations and Geometric Measure Theory

A. De Rosa - R. Tione

Regularity for graphs with bounded anisotropic mean curvature

created by derosa on 20 Nov 2020
modified on 02 Aug 2022

[BibTeX]

Published Paper

Inserted: 20 nov 2020
Last Updated: 2 aug 2022

Journal: Inventiones Mathematicae
Year: 2022

ArXiv: 2011.09922 PDF

Abstract:

We prove that $m$-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p>m$, are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a $C^{1,1}$ neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a $C^{1,1}$ (resp. $C^1$) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, for which the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs.


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