| Version 4 |
Version 3 |
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\newtheorem*{thm}{Theorem} |
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| \begin{thm}[Rado] |
\begin{thm}[Rado] |
| 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. |
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. |
| \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} |
