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

单词	RadosTheorem
释义	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
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