论文标题
具有界面各向异性平均曲率图的图表的规律性
Regularity for graphs with bounded anisotropic mean curvature
论文作者
论文摘要
我们证明，$ m $二维Lipschitz图具有各向异性平均曲率，以$ l^p $，$ p> m $界定，几乎在每个维度和编辑中几乎到处都是任何地方。这为文献中引起的多个开放问题提供了部分或完整的答案。需要各向异性能量来满足新型的椭圆形条件，例如，该条件在该区域功能的$ C^2 $附近。事实证明，这种情况暗示了原子状况。特别是，我们提供了在高的编码中满足原子状况的第一批非平凡的示例，以解决该领域的一个空旷问题。作为副产品，我们推断出具有$ c^2 $（分别$ c^1 $）的局部各向异性第一变化的Varifolds（Varifolds质量的分子）。除这些示例外，我们还提供了一类各向异性能量，远离面积功能，为此，具有局部界限各向异性第一变化的Varifolds的重构性。总而言之，我们表明，在各向异性固定图的情况下，原子条件不包括非平凡的年轻措施。
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 $C^2$ 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 $C^2$ (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.
