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)\leq u_{2}(z)\leq \dots

in every point of $G$ , then the limit

\lim _{n\to \infty }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 case the limits are finite, the limit function

u(z)=\lim _{n\to \infty }u_{n}(z)

is harmonic in $G$ .

References

Kamynin, L.I. (2001) [1994], "Harnack theorem", Encyclopedia of Mathematics, EMS Press