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This article is part of the series Harnack's Estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations.

Open Access Research Article

Harnack Inequalities: An Introduction

Moritz Kassmann

Author Affiliations

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

Boundary Value Problems 2007, 2007:081415  doi:10.1155/2007/81415


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/081415


Received:12 October 2006
Accepted:12 October 2006
Published:18 January 2007

© 2007 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.

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