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

TL;DR

This paper provides a geometric proof that all anisotropic minimal surfaces in the plane with strictly convex integrands are composed of line segments, establishing regularity and a Bernstein theorem for such surfaces.

Contribution

It offers a geometric proof confirming that strict convexity of the integrand ensures regularity of anisotropic minimal surfaces in 2, and proves a Bernstein theorem classifying global minimizers.

Findings

Anisotropic minimal surfaces in 2 are unions of line segments when the integrand is strictly convex.

Strict convexity of the integrand is both necessary and sufficient for regularity of minimal surfaces in 2.

Global minimizers are half-spaces, as established by the Bernstein theorem.

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.