Elliptic partial differential equations and quasiconformal mappings in the plane (princeton mathematical series)

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compellin...

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Hauptverfasser:	Astala, Kari, Iwaniec, Tadeusz, Martin, Gaven
Format:	E-Book Buch
Sprache:	Englisch
Veröffentlicht:	Princeton Princeton University Press 2009
Ausgabe:	1
Schriftenreihe:	Princeton Mathematical Series
Schlagworte:	Adjoint equation Analytic function Analytic proof Beltrami equation Boundary value problem Bounded mean oscillation Calculus of variations Cauchy–Riemann equations Central limit theorem Characterization (mathematics) Complex analysis Complex plane Conformal geometry Conjugate variables Continuous function (set theory) Degeneracy (mathematics) Differential equation Differential equations, Elliptic Dirichlet integral Dirichlet problem Distribution (mathematics) Elliptic operator Elliptic partial differential equation Equation Equations of motion Euler–Lagrange equation Explicit formulae (L-function) Factorization Fourier transform Fubini's theorem Geometric function theory Geometry Harmonic conjugate Harmonic function Hilbert transform Holomorphic function Homeomorphism Hyperbolic geometry Julia set Lagrangian (field theory) Laplace's equation Limit (mathematics) Linear differential equation Linear equation Linear fractional transformation Locally integrable function Lusin's theorem Mathematical optimization MATHEMATICS MATHEMATICS / Complex Analysis MATHEMATICS / Functional Analysis MATHEMATICS / Differential Equations / General Mirror symmetry (string theory) Moduli space Monodromy theorem Montel's theorem Operator (physics) Operator theory Partial differential equation Quadratic function Quasiconformal mapping Quasiconformal mappings Quasiconvex function Renormalization Riemann sphere Riemann surface Riemannian geometry Riesz–Thorin theorem Sign (mathematics) Sobolev space Square-integrable function Theorem Two-dimensional space Upper half-plane Variable (mathematics) Weyl's lemma (Laplace equation) Hyperbolic geometry Linear differential equation Monodromy theorem Laplace's equation Mirror symmetry (string theory) Partial differential equation Square-integrable function Hilbert transform Analytic proof Riesz–Thorin theorem Lagrangian (field theory) Distribution (mathematics) Degeneracy (mathematics) Characterization (mathematics) Two-dimensional space Julia set Explicit formulae (L-function) Complex plane Adjoint equation Linear fractional transformation Holomorphic function Fubini's theorem Riemann surface Conjugate variables Conformal geometry Harmonic function Boundary value problem Homeomorphism Lusin's theorem Mathematical optimization Beltrami equation Locally integrable function Quadratic function Bounded mean oscillation Continuous function (set theory) Sign (mathematics) Sobolev space Differential equation Complex analysis Harmonic conjugate Operator (physics) Dirichlet integral Fourier transform Cauchy–Riemann equations Operator theory Riemannian geometry Weyl's lemma (Laplace equation) Euler–Lagrange equation Linear equation Montel's theorem Moduli space Dirichlet problem Theorem Quasiconformal mapping Calculus of variations Upper half-plane Equation Analytic function Quasiconvex function Equations of motion Factorization Riemann sphere Geometric function theory Geometry Variable (mathematics) Central limit theorem Elliptic operator Renormalization Elliptic partial differential equation Limit (mathematics)
ISBN:	1400830117, 0691137773, 9780691137773, 9781400830114
Online-Zugang:	Volltext
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Beschreibung
Zusammenfassung:	This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.
Bibliographie:	Bibliography: p. 647-670 Includes index
ISBN:	1400830117 0691137773 9780691137773 9781400830114
DOI:	10.1515/9781400830114