A note on boundary point principles for partial differential inequalities of elliptic type

TL;DR

This paper explores boundary point principles for elliptic partial differential inequalities, emphasizing differences in conditions for maximum principles and boundary lemmas, and introduces new comparison and tangency principles with specific hypotheses.

Contribution

It introduces a boundary point lemma for quasi-linear elliptic inequalities and discusses the necessity of certain hypotheses, expanding understanding of boundary principles in elliptic PDEs.

Findings

Established a comparison-type boundary point lemma for elliptic solutions.

Highlighted the difference between conditions for maximum principles and boundary lemmas.

Demonstrated the necessity of hypotheses through simple examples.

Abstract

In this note we consider boundary point principles for partial differential inequalities of elliptic type. Firstly, we highlight the difference between conditions required to establish classical strong maximum principles and classical boundary point lemmas for second order linear elliptic partial differential inequalities. We highlight this difference by introducing a singular set in the domain where the coefficients of the partial differential inequality need not be defined, and in a neighborhood of which, can blow-up. Secondly, as a consequence, we establish a comparison-type boundary point lemma for classical elliptic solutions to quasi-linear partial differential inequalities. Thirdly, we consider tangency principles, for $C^1$ elliptic weak solutions to quasi-linear divergence structure partial differential inequalities. We highlight the necessity of certain hypotheses in the aforementioned results via simple examples.