A counterexample to the Hopf-Oleinik lemma (elliptic case)

TL;DR

This paper constructs a counterexample demonstrating that the Dini-type boundary condition is both necessary and sufficient for the Hopf-Oleinik boundary estimate in convex domains, confirming the condition's sharpness.

Contribution

It provides a new counterexample that confirms the sharpness of the Dini-type boundary condition for the Hopf-Oleinik lemma in convex domains.

Findings

Dini-type boundary condition is necessary and sufficient for Hopf-Oleinik estimates in convex domains.

Counterexample confirms the sharpness of the Dini-type condition.

The result clarifies boundary regularity requirements for elliptic PDE estimates.

Abstract

We construct a new counterexample confirming the sharpness of the Dini-type condition for the boundary of $\Omega$. In particular, we show that for convex domains the Dini-type assumption is the necessary and sufficient condition which guarantees the Hopf-Oleinik type estimates.