Fatou's Theorem and minimal graphs

TL;DR

This paper generalizes a result about minimal surface solutions in Euclidean discs to broader differential operators and constructs a minimal graph over a Hadamard surface with finite radial limits on a measure-zero set.

Contribution

It extends Collin-Rosenberg's theorem to divergence form operators and provides a new example of minimal graphs with specific boundary behavior on Hadamard surfaces.

Findings

Generalization of radial limit results to divergence form operators

Construction of minimal graphs with finite radial limits on measure-zero sets

Demonstration of boundary behavior in non-Euclidean geometries

Abstract

In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\mr$, where $\m$ is a Hadamard surface, over a geodesic disc which has finite radial limits in a mesure zero set.