Harnack's principle

Harnack's principle (Redirected from Harnack's theorem) Jump to navigation Jump to search 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 {estilo de exibição G} of the complex plane C, e {displaystyle u_{1}(z)leq u_{2}(z)leq dots } in every point of {estilo de exibição G} , then the limit {displaystyle lim _{até o infinito }você_{n}(z)} either is infinite in every point of the domain {estilo de exibição G} or it is finite in every point of the domain, in both cases uniformly in each compact subset of {estilo de exibição G} . In case the limits are finite, the limit function {estilo de exibição você(z)=lim_{até o infinito }você_{n}(z)} is harmonic in {estilo de exibição G} .

References Kamynin, L.I. (2001) [1994], "Harnack theorem", Enciclopédia de Matemática, EMS Press This article incorporates material from Harnack's principle on PlanetMath, que está licenciado sob a Licença Creative Commons Atribuição/Compartilhamento. Categorias: Harmonic functionsTheorems in complex analysisMathematical principles