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

Abstract

A set of locally finite perimeter $E \subset \mathbb{R}^{n}$ is called an anisotropic minimal surface in an open set $A$ if $\Phi(E;A) \le \Phi(F;A)$ for some surface energy $\Phi(E;A) = \int_{\partial^{*}E \cap A} \| \nu_{E}\| d \mathcal{H}^{n-1}$ and all sets of locally finite perimeter $F$ such that $E \Delta F \subset \subset A$. 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.