Rado’s theorem

Theorem (Rado).

Suppose $\mathrm{\Omega }\mathrm{\subset }{\mathrm{R}}^{\mathrm{2}}$ is a convex (http://planetmath.org/ConvexSet) domain (http://planetmath.org/Domain2) with a smooth boundary $\mathrm{\partial }\mathit{}\mathrm{\Omega }$ and suppose that $\mathrm{D}$ is the unit disc. Then given any homeomorphism $\mu \mathrm{:}\mathrm{\partial }\mathit{}\mathrm{D}\mathrm{\to }\mathrm{\partial }\mathit{}\mathrm{\Omega }$, there exists a unique harmonic function $u\mathrm{:}\mathrm{D}\mathrm{\to }\mathrm{\Omega }$ such that $u\mathrm{=}\mu$ on $\mathrm{\partial }\mathit{}\mathrm{D}$ and $u$ is a diffeomorphism.

Title	Rado’s theorem
Canonical name	RadosTheorem
Date of creation	2013-03-22 14:08:07
Last modified on	2013-03-22 14:08:07
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	8
Author	jirka (4157)
Entry type	Theorem
Classification	msc 31A05
Related topic	HarmonicFunction
Related topic	PerronFamily