Radó's theorem (harmonic functions) (original) (raw)
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 homeomorphismμ : ∂D → ∂Ω, there exists a unique harmonic function u : D → Ω such that u = μ on ∂D and u is a diffeomorphism.
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| dbo:abstract | 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 homeomorphismμ : ∂D → ∂Ω, there exists a unique harmonic function u : D → Ω such that u = μ on ∂D and u is a diffeomorphism. (en) En mathématiques, le théorème de Radó sur les fonctions harmoniques, nommé d'après Tibor Radó, exprime qu'une « bonne » forme « sans trous » peut être déformée de façon lisse en un disque. Soit Ω un ouvert convexe du plan euclidien R2 dont la frontière ∂Ω est lisse et soit D le disque unité ouvert. Alors, tout homéomorphismeμ : ∂ D → ∂ Ω se prolonge de façon unique en une fonction harmonique u : D → Ω. De plus, u est un difféomorphisme. (fr) |
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| rdfs:comment | 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 homeomorphismμ : ∂D → ∂Ω, there exists a unique harmonic function u : D → Ω such that u = μ on ∂D and u is a diffeomorphism. (en) En mathématiques, le théorème de Radó sur les fonctions harmoniques, nommé d'après Tibor Radó, exprime qu'une « bonne » forme « sans trous » peut être déformée de façon lisse en un disque. Soit Ω un ouvert convexe du plan euclidien R2 dont la frontière ∂Ω est lisse et soit D le disque unité ouvert. Alors, tout homéomorphismeμ : ∂ D → ∂ Ω se prolonge de façon unique en une fonction harmonique u : D → Ω. De plus, u est un difféomorphisme. (fr) |
| rdfs:label | Théorème de Radó (fonctions harmoniques) (fr) Radó's theorem (harmonic functions) (en) |
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