Partial results on extending the Hopf Lemma

TL;DR

This paper discusses extending a one-dimensional generalization of the Hopf Lemma to higher dimensions, proposing conjectures and providing partial insights to stimulate further research.

Contribution

It introduces two conjectures for higher-dimensional extensions of the Hopf Lemma and offers initial partial results, highlighting an open problem in mathematical analysis.

Findings

Proposed two conjectures for higher-dimensional Hopf Lemma extensions

Provided partial results supporting the conjectures

Called attention to an open problem in the field

Abstract

In [1], Theorem 3, the authors proved, in one dimension, a generalization of the Hopf Lemma, and the question arose if it could be extended to higher dimensions. In this paper we present two conjectures as possible extensions, and give a very partial answer. We write this paper to call attention to the problem.