Rado's theoremRadosTheorem2004-02-05 17:57:332006-06-21 11:13:19Theorem% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}\begin{thm}[Rado]
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.
\end{thm}
\begin{thebibliography}{9}
\bibitem{schoenyau}
R.\@ Schoen, S.\@ T.\@ Yau. \emph{\PMlinkescapetext{Lectures on Harmonic
Maps}}. International Press, Inc., Boston, Massachusetts, 1997
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