Rado's theorem

Rado's theorem

: "See also Rado's theorem (Ramsey theory)"

In mathematics, Rado's theorem is a result about harmonic functions. Informally, it says that any "nice looking" shape without holes can be smoothly deformed into a disk.

Suppose Ω is an open, connected and convex subset of the Euclidean space R2 with smooth boundary ∂Ω and suppose that D is the unit disk. Then, given any homeomorphism μ : ∂ D → ∂ Ω, there exists a unique harmonic function "u" : D → Ω such that "u" = μ on ∂D and "u" is a diffeomorphism.

References

* R. Schoen, S. T. Yau. (1997) Lectures on Harmonic Maps. International Press, Inc., Boston, Massachusetts. ISBN 1-57146-002-0.


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