Regularity for graphs with bounded anisotropic mean curvature

• 成果类型: Article
• 署名作者: De Rosa, Antonio; Tione, Riccardo
• 署名单位: University System of Maryland; University of Maryland College Park; Max Planck Society
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01129-6
• 发表日期: 2022
• 页码: 463-507
• 关键词: 1st variation; rectifiability; minimizers; calculus

摘要

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,C-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,C-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, forwhich 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.