A half-space Liouville theorem for anisotropic minimal graph with free boundary

TL;DR

This paper proves that anisotropic minimal graphs with free boundary in a half-space are necessarily flat if they grow at most linearly on one side, extending classical minimal surface results to free boundary problems.

Contribution

It extends classical Liouville theorems for minimal graphs to the setting of anisotropic free boundary problems with linear growth conditions.

Findings

Anisotropic minimal graphs with free boundary are flat under linear growth constraints.

The result generalizes classical minimal surface theorems to free boundary contexts.

The theorem applies to graphs with at most one-sided linear growth in the half-space.

Abstract

In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri-De Giorgi-Miranda and Simon to an appropriate free boundary setting.