A centennial of the Zaremba--Hopf--Oleinik Lemma

TL;DR

This paper explores the minimal conditions on lower-order coefficients in elliptic and parabolic equations necessary for the Hopf--Oleinik Lemma to hold, also addressing boundary gradient estimates.

Contribution

It provides new insights into the coefficient conditions ensuring the Hopf--Oleinik Lemma and boundary gradient estimates in second-order PDEs.

Findings

Identifies conditions on coefficients for the Hopf--Oleinik Lemma to be valid.

Establishes boundary gradient estimates under minimal assumptions.

Discusses the impact of coefficient 'badness' on solution behavior.

Abstract

We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure the Hopf--Oleinik Lemma for solutions to hold true. We also touch the gradient estimates for solutions at the boundary.