eBook

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

Iwaniec, Tadeusz, Astala, Kari, Martin, Gaven

2009

Overview

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.

• Publisher: Princeton University Press
• Subject: 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)
• ISBN: 1400830117, 0691137773, 9780691137773, 9781400830114