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 $\hat{I}¦(E;A)\le \hat{I}¦(F;A)$ for some surface energy $\hat{I}¦(E;A)={\int }_{{\partial }^{\ast }E\cap A}\Vert \hat{I}{½}_E\Vert d{\mathcal{H}}^{n-1}$ and all sets of locally finite perimeter $F$ such that $E\hat{I}”F\subset \subset A$. 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 $\Vert \cdot \Vert$ is strictly convex are locally disjoint unions of line segments. This demonstrates that, in the plane, strict convexity of $\Vert \cdot \Vert$ 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.