Harnack's principle
HarnacksPrinciple
2005-01-22 10:02:39
2006-11-23 21:19:01
Theorem
harmonic function
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If the functions\, $u_1(z)$, $u_2(z)$, \ldots\, are \PMlinkname{harmonic}{HarmonicFunction} in the domain\, $G \subseteq\mathbb{C}$\, and
$$u_1(z) \le u_2(z) \le \cdots$$
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 \PMlinkname{uniformly}{UniformConvergence} in each \PMlinkname{closed}{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).