Harnack’s principle

If the functions  $u_1(z)$, $u_2(z)$, …  are harmonic (http://planetmath.org/HarmonicFunction) in the domain  $G\subseteq ℂ$  and

$$u_1(z)\le u_2(z)\le \mathrm{\cdots }$$

in every point of $G$, then  ${lim}_{n\to \mathrm{\infty }}{u}_n(z)$  either is infinite in every point of the domain or it is finite in every point of the domain, in both cases uniformly (http://planetmath.org/UniformConvergence) in each closed (http://planetmath.org/ClosedSet) subdomain of $G$.  In the latter case, the function  $u(z)={lim}_{n\to \mathrm{\infty }}{u}_n(z)$  is harmonic in the domain $G$ (cf. limit function of sequence).

Title	Harnack’s principle
Canonical name	HarnacksPrinciple
Date of creation	2013-03-22 14:57:35
Last modified on	2013-03-22 14:57:35
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	14
Author	Mathprof (13753)
Entry type	Theorem
Classification	msc 30F15
Classification	msc 31A05