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- Beltrami_equation abstract "In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equationfor w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L2, and where μ is a given complex function in L∞(U) of norm less than 1, called the Beltrami coefficient. Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and relies on the Lp theory of the Beurling transform, a singular integral operator defined on LP(C) for all 1 < p < ∞. The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization theorem. The existence of conformal weldings can also be derived using the Beltrami equation. One of the simplest applications is to the Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equation can be used to show that the uniformizing map from the unit disk to the domain extends to a C∞ function from the closed disk to the closure of the domain.".
- Beltrami_equation wikiPageExternalLink books?id=8iZczAQc59cC.
- Beltrami_equation wikiPageExternalLink books?id=ZQjBXxxQsucC.
- Beltrami_equation wikiPageExternalLink TeichmullerVol1.html.
- Beltrami_equation wikiPageExternalLink beltrami01.pdf.
- Beltrami_equation wikiPageID "31356616".
- Beltrami_equation wikiPageRevisionID "602741783".
- Beltrami_equation hasPhotoCollection Beltrami_equation.
- Beltrami_equation subject Category:Complex_analysis.
- Beltrami_equation subject Category:Moduli_theory.
- Beltrami_equation subject Category:Operator_theory.
- Beltrami_equation subject Category:Partial_differential_equations.
- Beltrami_equation type Abstraction100002137.
- Beltrami_equation type Communication100033020.
- Beltrami_equation type DifferentialEquation106670521.
- Beltrami_equation type Equation106669864.
- Beltrami_equation type MathematicalStatement106732169.
- Beltrami_equation type Message106598915.
- Beltrami_equation type PartialDifferentialEquation106670866.
- Beltrami_equation type PartialDifferentialEquations.
- Beltrami_equation type Statement106722453.
- Beltrami_equation comment "In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equationfor w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L2, and where μ is a given complex function in L∞(U) of norm less than 1, called the Beltrami coefficient. Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric.".
- Beltrami_equation label "Beltrami equation".
- Beltrami_equation sameAs m.0gkym5_.
- Beltrami_equation sameAs Q4884794.
- Beltrami_equation sameAs Q4884794.
- Beltrami_equation sameAs Beltrami_equation.
- Beltrami_equation wasDerivedFrom Beltrami_equation?oldid=602741783.
- Beltrami_equation isPrimaryTopicOf Beltrami_equation.