On the Hopf Lemma

TL;DR

This paper extends the Hopf Lemma to domains with less regular boundaries for second order elliptic and parabolic operators, clarifying the boundary regularity conditions needed for the lemma to hold.

Contribution

It establishes the validity of the Hopf Lemma in domains with $C^{1,eta}$ and less regular boundaries, and identifies the boundary regularity threshold where it fails.

Findings

Hopf Lemma holds in $C^{1,eta}$ domains for elliptic operators

Hopf Lemma also valid for second order parabolic operators in similar domains

The lemma does not hold in $C^1$ boundary domains

Abstract

The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,\alpha}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.