Elliptic partial differential equations and quasiconformal mappings in the plane - Astala, Kari, 1953-; Iwaniec, Tadeusz; Martin, Gaven (Princeton, NJ : Princeton University Press, 2009)
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. Th.
Background in conformal geometry -- Foundations of quasiconformal mappings -- Complex potentials -- Measurable Riemann mapping theorem: the existence theory of quasiconformal mappings -- Parameterizing general linear elliptic systems -- Concept of ellipticity -- Solving general nonlinear first-order elliptic systems -- Nonlinear Riemann mapping theorems -- Conformal deformations and beltrami systems -- Quasilinear cauchy problem -- Holomorphic motions -- Higher Integrability -- L[superscript P]-theory of beltrami operators -- Schauder estimates for beltrami operators -- Applications to partial differential equations -- PDEs not of divergence type: pucci's conjecture -- Quasiconformal methods in impedance tomography: Calderón's problem -- Integral estimates for the Jacobian -- Solving the Beltrami equation: degenerate elliptic case -- Aspects of the calculus of variations -- Appendix. Elements of sobolev theory and function spaces -- A.1. Schwartz distributions -- A.2. Definitions of Sobolev spaces -- A.3. Mollification -- A.4. Pointwise coincidence of Sobolev functions -- A.5. Alternate characterizations -- A.6. Embedding theorems -- A.7. Duals and compact embeddings -- A.8. Hardy spaces and BMO -- A.9. Reverse holder inequalities -- A.10. Variations of Sobolev mappings.
Includes bibliographical references and index.
Print version record.
In English.
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Keywords: Differential equations, Elliptic; Quasiconformal mappings; Équations différentielles elliptiques; Applications quasi conformes; MATHEMATICS -- Complex Analysis; Differential equations, Elliptic; Quasiconformal mappings; Elliptische Differentialgleichung; Adjoint equation; Analytic function; Analytic proof; Banach space; Beltrami equation; Boundary value problem; Bounded mean oscillation; Calculus of variations; Cantor function; Cartesian product; Cauchy-Riemann equations; Central limit theorem; Characterization (mathematics).; Complex analysis; Complex plane; Conformal geometry; Conformal map; Conjugate variables; Continuous function (set theory).; Coordinate space; Degeneracy (mathematics).; Differential equation; Directional derivative; Dirichlet integral; Dirichlet problem; Disk (mathematics).; 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; Geometric measure theory; Geometry; Harmonic conjugate; Harmonic function; Harmonic map; Harmonic measure; Hilbert transform; Holomorphic function; Homeomorphism; Hyperbolic geometry; Hyperbolic trigonometry; Invertible matrix; Jacobian matrix and determinant; Julia set; Lagrangian (field theory).