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Harnack's theorem ：ウィキペディア英語版

Harnack's principle
In complex analysis, Harnack's principle or Harnack's theorem is one of several closely related theorems about the convergence of sequences of harmonic functions, that follow from Harnack's inequality.
If the functions

u_1(z)

u_2(z)

, ... are harmonic in an open connected subset

G

of the complex plane C, and
:

u_1(z) \le u_2(z) \le ...

in every point of

G

, then the limit
:

\lim_u_n(z)

either is infinite in every point of the domain

G

or it is finite in every point of the domain, in both cases uniformly in each compact subset of

G

. In the latter case, the function
:

u(z) = \lim_u_n(z)

is harmonic in the set

G

.
==References==

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抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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