# Rado’s theorem

###### Theorem (Rado).

Suppose $\Omega\subset{\mathbb{R}}^{2}$ is a convex (http://planetmath.org/ConvexSet) domain (http://planetmath.org/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.

## References

• 1 R. Schoen, S. T. Yau. . International Press, Inc., Boston, Massachusetts, 1997
Title Rado’s theorem RadosTheorem 2013-03-22 14:08:07 2013-03-22 14:08:07 jirka (4157) jirka (4157) 8 jirka (4157) Theorem msc 31A05 HarmonicFunction PerronFamily