Harnack's principle
The Harnack principle , also cited as Harnack's theorem , is a fundamental theorem from the mathematical branch of function theory , which goes back to the mathematician Axel Harnack (1851–1888), who presented this theorem in a paper from 1886. Harnack's 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 classic complex case
Given is an open set and a sequence of harmonic functions , which increase monotonically at points:
- .
Be for
Be on
and
Then:
- (1) Both and are at the same time open and closed in .
- (2) In the event that an area is from , either always applies to or always applies to .
- (3) If a domain of and holds for a , then the sequence of functions converges locally uniformly and the limit function is also a harmonic function.
Generalization to higher dimensions
As Axel Harnack himself suggests, the corresponding principle with a very similar formulation also applies to the case of harmonic functions on open sets of . Here the proof is based on the n-dimensional version of Harnack's inequality.
literature
Original work
- Axel Harnack: Evidence of existence for the theory of potential in the plane and in space . In: Reports on the negotiations of the Royal Saxon Society of Sciences . 1886, p. 144-169 .
- Axel Harnack: Evidence of existence for the theory of potential in the plane and in space . In: Mathematical Annals . tape 35 , 1890, pp. 19-40 .
Monographs
- Lars Valerian Ahlfors : Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable . 3. Edition. McGraw-Hill, New York [et. a.] 1979, ISBN 0-07-000657-1 .
- Sheldon Axler, Paul Bourdon, Wade Ramey: Harmonic Function Theory . Springer-Verlag, Berlin [a. a.] 1992, ISBN 3-540-97875-5 .
- Eberhard Freitag : Function Theory 2 (= Springer textbook ). Springer-Verlag, Berlin [a. a.] 2009, ISBN 978-3-540-87899-5 .
- WK Hayman, PB Kennedy: Subharmonic functions (= LMS Monographs . Volume 9 ). Volume I. Academic Press, London [u. a.] 1976.
- Rolf Nevanlinna , Veikko Paatero: Introduction to the theory of functions (= textbooks and monographs from the field of exact sciences: Mathematical series . Volume 30 ). Birkhäuser Verlag, Basel / Stuttgart 1965.
- Walter Rudin : Real and Complex Analysis . 2nd improved edition. Oldenbourg Wissenschaftsverlag, Munich 2009, ISBN 978-3-486-59186-6 .
Individual evidence
- ↑ Harnack: Ber. Negotiating Kings Saxon. Companion Knowledge Leipzig . 1886, p. 144 ff .
- ↑ Friday: p. 59 ff.
- ↑ Nevanlinna / Paatero: p. 234 ff.
- ↑ Rudin: p. 283 ff.
- ↑ See concluding remark in his treatise in the Math. Ann., Volume 35, p. 40.
- ↑ Hayman / Kennedy: pp. 35 ff.
- ↑ Axler / Bourdon / Ramey: p. 47 ff.