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: ${\displaystyle D\subset \mathbb {C} }$ ${\displaystyle (u_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {u_{n}}\colon \,D\to \mathbb {R} }$ ${\displaystyle (n\in \mathbb {N} )}$

${\displaystyle u_{1}(z)\leq u_{2}(z)\leq \ldots }$ ${\displaystyle (z\in D)}$.

Be for ${\displaystyle z\in D}$

${\displaystyle u(z)=\sup _{n\in \mathbb {N} }{u_{n}(z)}=\lim _{n\to \infty }{u_{n}(z)}}$

Be on

${\displaystyle D_{0}=\{z\in D:\,u(z)<\infty \}}$

and

${\displaystyle D_{1}=\{z\in D:\,u(z)=\infty \}}$

Then:

(1) Both and are at the same time open and closed in . ${\displaystyle D_{0}}$ ${\displaystyle D_{1}}$ ${\displaystyle D}$

(2) In the event that an area is from , either always applies to or always applies to . ${\displaystyle D}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle u(z)=\infty }$ ${\displaystyle z\in D}$ ${\displaystyle u(z)<\infty }$ ${\displaystyle z\in D}$

(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. ${\displaystyle D}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle u(z_{0})<\infty }$ ${\displaystyle z_{0}\in D}$ ${\displaystyle u}$

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. ${\displaystyle {\mathbb {R} }^{n}}$

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

1. ↑ Harnack: Ber. Negotiating Kings Saxon. Companion Knowledge Leipzig . 1886, p. 144 ff .
2. ↑ Friday: p. 59 ff.
3. ↑ Nevanlinna / Paatero: p. 234 ff.
4. ↑ Rudin: p. 283 ff.
5. ↑ See concluding remark in his treatise in the Math. Ann., Volume 35, p. 40.
6. ↑ Hayman / Kennedy: pp. 35 ff.
7. ↑ Axler / Bourdon / Ramey: p. 47 ff.