Harnack's principle

The Harnack principle, also cited as a set of Harnack, is a basic set from the mathematical branch of function theory, which the mathematician Axel Harnack (1851-1888) back, who has argued that sentence in a work of 1886. The Harnack principle deals with the convergence behavior of monotonically increasing sequences of harmonic functions. It is based on the inequality also found by Axel Harnack and named after him.

Formulation of the principle in the classical complex case

Given an open set and to a series of harmonic functions which anwachse pointwise monotone:

Be for

Be more

And

Then:

Generalization to higher dimensions

As Axel Harnack suggests itself, the corresponding principle for very similar formulation applies also for the case of harmonic functions on open sets. Here the proof is based on the n- dimensional version of the Harnack inequality.

Swell

Original Article

• Axel Harnack: existence proofs for the theory of the potential in the plane and in space. In: Ber. Verhandl. Kings. Saxon. Gesell. Wiss. Leipzig. 1886, pp. 144-169.

Monographs

• Lars Valerian Ahlfors: Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable. 3rd edition. McGraw- Hill, New York [ua ] 1979, ISBN 0-07-000657-1.
• Sheldon Axler; Paul Bourdon; Wade Ramey: Harmonic Function Theory. Springer -Verlag, Berlin [ ua] 1992, ISBN 3-540-97875-5.
• Eberhard Friday: Function Theory 2 ( = Springer - Lehrbuch ). Springer -Verlag, Berlin [ ua] 2009, ISBN 978-3-540-87899-5.
• WK Hayman and PB Kennedy, Subharmonic functions. Volume I (= L. M. S. Monographs. 9). Academic Press, London [ et al ] 1976.
• Rolf Nevanlinna and Paatero Veikko: Introduction to the theory of functions ( = textbooks and monographs in the field of exact sciences: Mathematical number 30. ). Birkhauser Verlag, Basel and Stuttgart, 1965.
• Walter Rudin: Real and Complex Analysis. 2nd revised edition. Oldenbourg Verlag, Munich 2009, ISBN 978-3-486-59186-6.