PlanetMath: Rado's theorem

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Rado's theorem

(Theorem)

Theorem 1 (Rado) Suppose $\Omega \subset {\mathbb{R}}^2$ is a convex domain with a smooth boundary $\partial \Omega$ and suppose that ${\mathbb{D}}$ is the unit disc. Then given any homeomorphism $\mu : \partial {\mathbb{D}} \rightarrow \partial \Omega$ , there exists a unique harmonic function $u : {\mathbb{D}} \rightarrow \Omega$ such that $u = \mu$ on $\partial {\mathbb{D}}$ and is a diffeomorphism.

Bibliography

1: R. Schoen, S. T. Yau. Lectures on Harmonic Maps. International Press, Inc., Boston, Massachusetts, 1997

"Rado's theorem" is owned by jirka.

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See Also: harmonic function, Perron family

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Cross-references: diffeomorphism, harmonic function, homeomorphism, unit disc, boundary, smooth
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This is version 5 of Rado's theorem, born on 2004-02-05, modified 2006-06-21.
Object id is 5549, canonical name is RadosTheorem.
Accessed 1743 times total.

Classification:

AMS MSC:

31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions)

Pending Errata and Addenda

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