Boundary Value Problems
Volume 2007 (2007), Article ID 81415, 21 pages
Research Article

Harnack Inequalities: An Introduction

Moritz Kassmann

Institute of Applied Mathematics, University of Bonn, Beringstrasse 6, Bonn 53115, Germany

Received 12 October 2006; Accepted 12 October 2006

Academic Editor: Ugo Pietro Gianazza

Copyright © 2007 Moritz Kassmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The aim of this article is to give an introduction to certain inequalities named after Carl Gustav Axel von Harnack. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the general theory of harmonic functions and partial differential equations. We restrict ourselves mainly to the analytic perspective but comment on the geometric and probabilistic significance of Harnack inequalities. Our focus is on classical results rather than latest developments. We give many references to this topic but emphasize that neither the mathematical story of Harnack inequalities nor the list of references given here is complete.