Harnack Inequalities: An Introduction

By M. Kassmann. 2007

66 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/75e47c2f7d2affdb9b808aad0b754d27ce1f8522

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Harnack inequality for singular fully nonlinear operators and some existence results (2010, methodology): “We refer the reader to the work by Kassmann [29] for a very complete bibliographic review on Harnack inequality.”
Regularity estimates for elliptic nonlocal operators (2015, methodology): “The second author would like to use the opportunity to correct an error in [19] concerning the name Harnack.”
Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms (2016, background): “ty (IRTSTFJ), and the JSPS postdoctoral fellowship (26·04021). 1 (at least for the cases of local operators/diﬀusions) they imply H¨older continuity of harmonic/caloric functions. We refer readers to [K1] for the history and the basic introduction of Harnack inequalities. Because of their fundamenta”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, methodology): “Sin embargo, una f ormula abstracta no es util a la hora de tratar problemas concretos de EDPs y se necesitar a una expresi on puntual m as o menos expl cita para Lσf(x).”