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 ${\displaystyle u_{1}(z)}$, ${\displaystyle u_{2}(z)}$, ... are harmonic in an open connected subset ${\displaystyle G}$ of the complex plane C, and

${\displaystyle u_{1}(z)\leq u_{2}(z)\leq \dots }$

in every point of ${\displaystyle G}$, then the limit

${\displaystyle \lim _{n\to \infty }u_{n}(z)}$

either is infinite in every point of the domain ${\displaystyle G}$ or it is finite in every point of the domain, in both cases uniformly in each compact subset of ${\displaystyle G}$. In case the limits are finite, the limit function

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

is harmonic in ${\displaystyle G}$.

References

• Kamynin, L.I. (2001) [1994], "Harnack theorem", Encyclopedia of Mathematics, EMS Press
• This article incorporates material from Harnack's principle on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.