arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2007.12953math.DGmath.APmath.OC
收藏

$\mathbb{R}^{2}$中所有各向异性极小曲面正则性的几何证明

A geometric proof of regularity of all anisotropic minimal surfaces in $\mathbb{R}^{2}$

Max Goering

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

一个局部有限周长$E \subset \mathbb{R}^{n}$的集合称为开集$A$中的各向异性极小曲面，如果对于某个表面能$Φ(E;A) = \int_{\partial^{*}E \cap A} \| ν_{E}\| d \mathcal{H}^{n-1}$和所有局部有限周长$F$的集合都满足$Φ(E;A) \le Φ(F;A)$，使得$E ΔF \subset \subset A$ .在这个简短的说明中，我们提供了一个几何证明的细节，验证所有各向异性表面极小的$\mathbb{R}^{2}$，其相应的被积函数$\| \cdot \|$是严格凸的局部不相交的工会的线段。这表明，在平面上，$\| \cdot \|$的严格凸性是正则性的充分必要条件.证明了相应的Bernstein定理：全局各向异性极小元$E \subset \mathbb{R}^{2}$是半空间.

英文摘要：

A set of locally finite perimeter $E \subset \mathbb{R}^{n}$ is called an anisotropic minimal surface in an open set $A$ if $Φ(E;A) \le Φ(F;A)$ for some surface energy $Φ(E;A) = \int_{\partial^{*}E \cap A} \| ν_{E}\| d \mathcal{H}^{n-1}$ and all sets of locally finite perimeter $F$ such that $E ΔF \subset \subset A$. In this short note we provide the details of a geometric proof verifying that all anisotropic surface minimizers in $\mathbb{R}^{2}$ whose corresponding integrand $\| \cdot \|$ is strictly convex are locally disjoint unions of line segments. This demonstrates that, in the plane, strict convexity of $\| \cdot \|$ is both necessary and sufficient for regularity. The corresponding Bernstein theorem is also proven: global anisotropic minimizers $E \subset \mathbb{R}^{2}$ are half-spaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑