Radó's theorem (harmonic functions)

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See also Rado's theorem (Ramsey theory)

In mathematics, Radó's theorem is a result about harmonic functions, named after Tibor Radó. 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.


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

This article incorporates material from Rado's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.