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Revision difference : Rado's theorem |
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Version 4 |
| \begin{thm}[Rado] |
\begin{thm}[Rado] |
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Suppose $\Omega \subset {\mathbb{R}}^2$ is a \PMlinkname{convex}{ConvexSet} \PMlinkname{domain}{Domain2} 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 $u$ is a diffeomorphism.
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Suppose $\Omega \subset {\mathbb{R}}^2$ is a \PMlinkname{convex}{ConvexSet} 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 $u$ is a diffeomorphism.
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| \end{thm} |
\end{thm} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{schoenyau} |
\bibitem{schoenyau} |
| R.\@ Schoen, S.\@ T.\@ Yau. \emph{\PMlinkescapetext{Lectures on Harmonic |
R.\@ Schoen, S.\@ T.\@ Yau. \emph{\PMlinkescapetext{Lectures on Harmonic |
| Maps}}. International Press, Inc., Boston, Massachusetts, 1997 |
Maps}}. International Press, Inc., Boston, Massachusetts, 1997 |
| \end{thebibliography} |
\end{thebibliography} |
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