About: Beltrami equation

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In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation for 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, and where and are Wirtinger derivatives. 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 < ∞.

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rdf:type	Thing yago:WikicatPartialDifferentialEquations yago:Abstraction100002137 yago:Communication100033020 yago:DifferentialEquation106670521 yago:Equation106669864 yago:MathematicalStatement106732169 yago:Message106598915 yago:PartialDifferentialEquation106670866 yago:Statement106722453
rdfs:label	Beltrami equation (en)
rdfs:comment	In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation for 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, and where and are Wirtinger derivatives. 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 < ∞. (en)
differentFrom	Laplace–Beltrami equation
rdfs:seeAlso	Conformal welding Simultaneous uniformization theorem Riemann mapping theorem
dcterms:subject	Moduli theory Operator theory Complex analysis Partial differential equations
Wikipage page ID	31356616 (xsd:integer)
Wikipage revision ID	1117730484 (xsd:integer)
Link from a Wikipage to another Wikipage	Quasiconformal mapping Moduli theory Holomorphic function Riemann surface Riesz–Thorin theorem Douady–Earle extension Limit set Operator theory Mathematics Wirtinger derivatives Equicontinuous Fundamental solution Conformal welding Cross-ratio Lipman Bers Singular integral operators of convolution type Sobolev space Fundamental group Journal d'Analyse Mathématique Measurable Riemann mapping theorem Adrien Douady American Mathematical Society Eugenio Beltrami First fundamental form Partial differential equation Quasisymmetric map Riemannian manifold Isothermal coordinates Jacobian matrix and determinant Plane curve Arzelà–Ascoli theorem Complex analysis Partial differential equations Jordan curve Borel's lemma Bulletin of the AMS Bump function Fredholm operator Free group Schoenflies theorem Schwartz function Hölder continuous Radius of curvature (mathematics) Schottky group Uniformization theorem Vanishing mean oscillation Neumann series Compact Riemann surface Simultaneous uniformization theorem Poincaré metric Quasi-Fuchsian group Weyl's lemma (Laplace equation) Unit disk Riemann mapping theorem Universal covering space Upper half plane Hölder inequality Beurling transform Elliptic regularity Covering map Gauss Teichmüller theory Alexander trick Singular integral operator Complex variable
Link from a Wikipage to an external page	http://matrixeditions.com/TeichmullerVol1.html http://www.caressa.it/pdf/beltrami01.pdf https://books.google.com/books%3Fid=8iZczAQc59cC https://books.google.com/books%3Fid=ZQjBXxxQsucC http://projecteuclid.org/euclid.bams/1183523321
sameAs	Beltrami equation Beltrami equation Beltrami equation Beltrami equation
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