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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.

Differential equations, Elliptic. --- Quasiconformal mappings. --- Mappings, Quasiconformal --- Conformal mapping --- Functions of complex variables --- Geometric function theory --- Mappings (Mathematics) --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial --- 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). --- Laplace's equation. --- Limit (mathematics). --- Linear differential equation. --- Linear equation. --- Linear fractional transformation. --- Linear map. --- Linearization. --- Lipschitz continuity. --- Locally integrable function. --- Lusin's theorem. --- Mathematical optimization. --- Mathematics. --- Maxima and minima. --- Maxwell's equations. --- Measure (mathematics). --- Metric space. --- Mirror symmetry (string theory). --- Moduli space. --- Modulus of continuity. --- Monodromy theorem. --- Monotonic function. --- Montel's theorem. --- Operator (physics). --- Operator theory. --- Partial derivative. --- Partial differential equation. --- Poisson formula. --- Polynomial. --- Quadratic function. --- Quasiconformal mapping. --- Quasiconvex function. --- Quasisymmetric function. --- Renormalization. --- Riemann sphere. --- Riemann surface. --- Riemannian geometry. --- Riesz transform. --- Riesz–Thorin theorem. --- Sign (mathematics). --- Sobolev space. --- Square-integrable function. --- Support (mathematics). --- Theorem. --- Two-dimensional space. --- Uniformization theorem. --- Upper half-plane. --- Variable (mathematics). --- Weyl's lemma (Laplace equation).