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 {stile di visualizzazione G} of the complex plane C, e {displaystyle u_{1}(z)leq u_{2}(z)leq dots } in every point of {stile di visualizzazione G} , then the limit {displaystyle lim _{infty }tu_{n}(z)} either is infinite in every point of the domain {stile di visualizzazione G} or it is finite in every point of the domain, in both cases uniformly in each compact subset of {stile di visualizzazione G} . In case the limits are finite, the limit function {stile di visualizzazione u(z)=lim _{infty }tu_{n}(z)} is harmonic in {stile di visualizzazione G} .

References Kamynin, L.I. (2001) [1994], "Harnack theorem", Enciclopedia della matematica, EMS Press This article incorporates material from Harnack's principle on PlanetMath, che è concesso in licenza in base alla licenza Creative Commons Attribution/Share-Alike. Categorie: Harmonic functionsTheorems in complex analysisMathematical principles