Geometric Function Theory and Non-linear Analysis

Hardcover | October 11, 2001

byTadeusz Iwaniec, Gaven John Martin

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This book provides a survey of recent developments in the field of non-linear analysis and the geometry of mappings.Sobolev mappings, quasiconformal mappings, or deformations, between subsets of Euclidean space, or manifolds or more general geometric objects may arise as the solutions to certain optimisation problems in the calculus of variations or in non-linear elasticity, as the solutions to differentialequations (particularly in conformal geometry), as local co-ordinates on a manifold or as geometric realisations of abstract isomorphisms between spaces such as those that arise in dynamical systems (for instance in holomorphic dynamics and Kleinian groups). In each case the regularity andgeometric properties of these mappings and related non-linear quantities such as Jacobians, tells something about the problems and the spaces under consideration.The applications studied include aspects of harmonic analysis, elliptic PDE theory, differential geometry, the calculus of variations as well as complex dynamics and other areas. Indeed it is the strong interactions between these areas and the geometry of mappings that underscores and motivates theauthors' work. Much recent work is included. Even in the classical setting of the Beltrami equation or measurable Riemann mapping theorem, which plays a central role in holomorphic dynamics, Teichmuller theory and low dimensional topology and geometry, the authors present precise results in thedegenerate elliptic setting. The governing equations of non-linear elasticity and quasiconformal geometry are studied intensively in the degenerate elliptic setting, and there are suggestions for potential applications for researchers in other areas.

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This book provides a survey of recent developments in the field of non-linear analysis and the geometry of mappings.Sobolev mappings, quasiconformal mappings, or deformations, between subsets of Euclidean space, or manifolds or more general geometric objects may arise as the solutions to certain optimisation problems in the calculus of...

Tadeusz Iwaniec is a John Raymond French Distinguished Professor of Mathematics at Syracuse University. Gaven John Martin is a Professor of Mathematics and James Cook Fellow of Royal Society, New Zealand.
Format:HardcoverDimensions:568 pages, 9.21 × 6.14 × 1.34 inPublished:October 11, 2001Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198509294

ISBN - 13:9780198509295

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Table of Contents

0. Introduction and Overview1. Conformal Mappings2. Stability of the Mobius Group3. Sobolev Theory and Function Spaces4. The Liouville Theorem5. Mappings of Finite Distortion6. Continuity7. Compactness8. Topics from Multilinear Algebra9. Differential Forms10. Beltrami Equations11. Riesz Transforms12. Integral Estimates13. The Gehring Lemma14. The Governing Equations15. Topological Properties of Mappings of Bounded Distortion16. Painleve's Theorem in Space17. Even Dimensions18. Picard and Montel Theorems in Space19. Conformal Structures20. Uniformly Quasiregular Mappings21. Quasiconformal Groups22. Analytic Continuation for Beltrami Systems