# Harnack theorem

Harnack's first theorem: If a sequence of functions which are harmonic in a bounded domain and continuous on converges uniformly on the boundary , then it also converges uniformly in to a harmonic function. This theorem can be generalized to solutions of an elliptic equation,

 (*)

which has a unique solution of the Dirichlet problem for any continuous boundary function . If the sequence of solutions of equation (*) converges uniformly on , then it also converges uniformly in to a solution of equation (*).

Harnack's second theorem, the Harnack principle: If a monotone sequence of harmonic functions in a bounded domain converges at some point in , then it converges at all points of to a harmonic function, and this convergence is uniform on any closed subdomain of . Harnack's second theorem can be generalized to monotone sequences of solutions of the elliptic equation .

#### References

 [1] I.G. [I.G. Petrovskii] Petrowski, "Vorlesungen über partielle Differentialgleichungen" , Teubner (1965) (Translated from Russian) [2] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)