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Boundary Value Problems

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  • Research Article
  • Open Access

Harnack Inequalities: An Introduction

Boundary Value Problems20072007:081415

https://doi.org/10.1155/2007/81415

©  Moritz Kassmann. 2007

Received: 12 October 2006

Accepted: 12 October 2006

Published: 18 January 2007

Abstract

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.

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • General Theory
  • Functional Equation

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Authors’ Affiliations

(1)
Institute of Applied Mathematics, University of Bonn, Bonn, Germany

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© Moritz Kassmann. 2007

This article is published under license to BioMed Central Ltd. 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.

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