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 <math> u_1(z)</math>, <math> u_2(z)</math>, ... are harmonic in an open subset <math>G</math> of the complex plane C, and <math>u_1(z) \le u_2(z) \le ...

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  • 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 <math> u_1(z)</math>, <math> u_2(z)</math>, ... are harmonic in an open subset <math>G</math> of the complex plane C, and <math>u_1(z) \le u_2(z) \le ... </math> in every point of <math>G</math>, then the limit <math> \lim_{n\to\infty}u_n(z)</math> either is infinite in every point of the domain <math>G</math> or it is finite in every point of the domain, in both cases uniformly in each compact subset of <math>G</math>. In the latter case, the function <math> u(z) = \lim_{n\to\infty}u_n(z)</math> is harmonic in the set <math> G</math>.
  • En analyse complexe, le principe de Harnack est un théorème concernant la convergence de fonctions harmoniques. Si les fonctions <math> u_1(z)</math>, <math> u_2(z)</math>, ... sont harmoniques sur l(ouvert <math>G</math> du plan complexe C, et <math>u_1 \le u_2 \le ... </math> en tout point de <math>G</math>, alors la limite <math> \lim_{n\to\infty}u_n</math> est soit infinie en chaque point du domaine de définition <math>G</math>, soit finie en chaque point de ce domaine. Dans tous les cas, la congence est uniforme sur chaque sous-ensemble compact de <math>G</math>. Dans le dernier cas, la fonction <math> u = \lim_{n\to\infty}u_n</math> est harmonique sur <math> G</math>.
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  • L.I.
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  • Kamynin
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  • Harnack theorem
  • Harnack's principle
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  • 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 <math> u_1(z)</math>, <math> u_2(z)</math>, ... are harmonic in an open subset <math>G</math> of the complex plane C, and <math>u_1(z) \le u_2(z) \le ...
  • En analyse complexe, le principe de Harnack est un théorème concernant la convergence de fonctions harmoniques. Si les fonctions <math> u_1(z)</math>, <math> u_2(z)</math>, ... sont harmoniques sur l(ouvert <math>G</math> du plan complexe C, et <math>u_1 \le u_2 \le ...
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  • Harnack's principle
  • Principe de Harnack
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