# Harnack’s principle

If the functions$u_{1}(z)$, $u_{2}(z)$, …  are harmonic (http://planetmath.org/HarmonicFunction) in the domain  $G\subseteq\mathbb{C}$  and

 $u_{1}(z)\leq u_{2}(z)\leq\cdots$

in every point of $G$, then  $\lim_{n\to\infty}u_{n}(z)$  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  $u(z)=\lim_{n\to\infty}u_{n}(z)$  is harmonic in the domain $G$ (cf. limit function of sequence).

Title Harnack’s principle HarnacksPrinciple 2013-03-22 14:57:35 2013-03-22 14:57:35 Mathprof (13753) Mathprof (13753) 14 Mathprof (13753) Theorem msc 30F15 msc 31A05