论文标题
$ \ mathbb {r}^{2} $的所有各向异性最小表面的规律性证明
A geometric proof of regularity of all anisotropic minimal surfaces in $\mathbb{R}^{2}$
论文作者
论文摘要
一组本地有限的周长$ e \ subset \ mathbb {r}^{n} $称为开放集中的各向异性最小表面$ a $ a $ a $（e; a; a）\leφ（f; a）$，用于某些表面能量$ $ $ $ $ $ $ ug ν_{E} \ | d \ Mathcal {h}^{n-1} $和所有局部有限的周长$ f $，以便$eΔf\ subset \ subset a $。 在此简短说明中，我们提供了几何证明的详细信息，以验证所有各向异性表面最小化器中的所有各向异性表面最小化器中的所有各向异性表面{r}^{2} $，其对应的intement $ \ | \ cdot \ | $严格凸出是线段的本地不相交工会。这表明，在飞机上，$ \ |的严格凸度。 \ cdot \ | $对于规律性既需要且足够。相应的Bernstein定理也已被证明：全局各向异性最小化器$ e \ subset \ mathbb {r}^{2} $是半空间。
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.
