Vol. 9, No. 2, 2016

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

Darya E. Apushkinskaya and Alexander I. Nazarov

Vol. 9 (2016), No. 2, 439–458
Abstract

We construct a new counterexample to the Hopf–Oleinik boundary point lemma. It shows that for convex domains, the ${C}^{1,Dini}$ assumption on $\partial \Omega$ is the necessary and sufficient condition providing the estimates of Hopf–Oleinik type.

 Dedicated to Professor M.V. Safonov
Keywords
elliptic equations, Hopf–Oleinik lemma, Dini continuity, counterexample
Mathematical Subject Classification 2010
Primary: 35J15, 35B45
Milestones
Revised: 29 September 2015
Accepted: 16 December 2015
Published: 24 March 2016
Authors
 Darya E. Apushkinskaya Department of Mathematics Saarland University P.O. Box 151150 Saarbrücken 66041 Germany St. Petersburg State University 7/9 Universitetskaya nab. St. Petersburg 199034 Russia Alexander I. Nazarov St. Petersburg Department Steklov Institute Fontanka 27 St. Petersburg 191023 Russia St. Petersburg State University 7/9 Universitetskaya nab. St. Petersburg 199034 Russia