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The Beltrami Equation : A Geometric Approach / by Vladimir Gutlyanskii, Vladimir Ryazanov, Uri Srebro, Eduard Yakubov
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Developments in Mathematics ; 26 | SpringerLink BücherPublisher: New York, NY : Springer New York, 2012Description: Online-RessourceISBN:- 9781461431916
- 1280802758
- 9781280802751
- 515.9 515/.9
- 515.353
- QA360
Contents:
Summary: 1. Introduction -- 2. Preliminaries -- 3. The Classical Beltrami Equation ||μ||∞ < 1 -- 4. The Degenerate Case -- 5. BMO- and FMO-Quasiconformal Mappings -- 6. Ring Q-Homeomorphisms at Boundary Points -- 7. Strong Ring Solutions of Beltrami Equations -- 8. On the Dirichlet Problem for Beltrami Equations -- 9. On the Beltrami Equations with Two Characteristics -- 10. Alternating Beltrami Equation -- References -- Index.Summary: The Beltrami Equation: A Geometric Approach will be particularly useful to many specialists in modern geometric analysis, quasiconformal mappings and extensions, beginning researchers, and graduate students with a year’s background in complex variables. This book covers the state-of-the art in the ongoing study of the Beltrami equation, the classical equation that has been studied for more than 100 years. Along with its rich history, the Beltrami equation plays a significant role in geometry, analysis, and physics. The most important feature of this work concerns the unified geometric approach taken based on the modulus method that can be effectively applied to solving many problems in mathematical physics. Beautiful examples illustrate the relationship between mappings with bounded oscillation and those with finite oscillations. Written by authors that are well-known specialists in this field, this monograph presents recent developments in the theory of Beltrami equations, studying a variety of problems like convergence, existence, uniqueness, representation, removal of singularities, local distortion estimates, and boundary behavior of solutions to the Beltrami equations. It contains new types of criteria in the given problems, particularly new integral conditions for the existence of regular solutions to the Beltrami equations that turned out to be not only sufficient but also necessary.PPN: PPN: 1651475229Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
The Beltrami Equation; Preface; Contents; Chapter1 Introduction; 1.1 The Beltrami Equation; 1.2 Historical Remarks; 1.3 Applications of Beltrami Equations; 1.4 Classification of Beltrami Equations; 1.5 ACL Solutions; 1.6 Ellipticity of the Beltrami Equation; Chapter2 Preliminaries; 2.1 BMO Functions in C; 2.2 BMO Functions in C; 2.2.1 Removability of Isolated Singularities of BMO Functions; 2.2.2 BMO Functions, qc Mappings, qc Arc, and Symmetric Extensions; 2.3 FMO Functions; 2.3.1 Examples of Functions FMOBMOloc; 2.4 On Sobolev's Classes; 2.5 Modulus and Capacity
2.6 Convergence Theorems for Homeomorphisms2.7 Ring Q-Homeomorphisms at Inner Points; 2.8 On Some Equivalent Integral Conditions; 2.9 One More Integral Condition; 2.10 On Weakly Flat and Strongly Accessible Boundaries; Chapter 3 The Classical Beltrami Equation ||u||<1; 3.1 Quasiconformal Mappings; 3.2 The Main Problems; 3.3 Integrability; 3.4 The Classical Existence and Uniqueness Theorem; 3.5 Methods of Proof of Uniqueness and Existence; 3.5.1 Uniqueness; 3.5.2 Existence; 3.5.2.1 Local Solutions; 3.5.2.2 Global Solutions via Local Solutions; 3.5.2.3 Global Solutions Directly
3.5.2.4 PDE Methods3.5.2.5 Singular Integral Methods; 3.5.3 Smoothness of the Solutions; 3.5.4 Analytic Dependence on Parameters; Chapter4 The Degenerate Case; 4.1 Examples; 4.1.1 Example One; 4.1.2 Example Two; 4.2 The Singular Set; 4.3 Auxiliary Results; 4.4 Case (i): The Singular Set E is Specified and E D; 4.4.1 Existence and Uniqueness; 4.4.2 Boundary Behavior; 4.4.3 Proof of Theorem 4.1; 4.4.4 Proof of Theorem 4.2; 4.4.5 Examples; 4.4.5.1 Example 1; 4.4.5.2 Example 2; 4.4.5.3 Example 3; 4.4.5.4 Example 4; 4.5 Case (ii): The Singular Set E is Specified and E D
4.6 Case (iii): The Singular Set E Is Not Specified4.6.1 Pesin; 4.6.2 Miklyukov and Suvorov; 4.6.3 Lehto; 4.6.4 Brakalova and Jenkins; 4.6.5 Iwaniec and Martin; 4.6.6 Gutlyanskii, Martio, Sugawa, and Vuorinen; 4.6.7 David; 4.6.8 Tukia; 4.6.9 Ryazanov, Srebro and Yakubov; 4.7 Modulus Inequalities; Chapter5 BMO- and FMO-Quasiconformal Mappings; 5.1 Introduction; 5.2 Inclusions, Integrability, and Group Properties; 5.3 Distortion Lemmas; 5.4 One Existence Theorem; 5.5 Uniqueness and Approximation; 5.5.1 Good Approximation; 5.6 Other Properties; 5.7 The Main Lemma on FMO
5.8 Estimate of Distortion5.9 Further Existence Theorems; Chapter6 Ring Q-Homeomorphisms at Boundary Points; 6.1 Introduction; 6.2 Examples and Properties; 6.3 The Completeness of Ring Homeomorphisms; 6.4 One Integral Inequality; 6.5 Distortion Estimates; 6.6 On Removability of Isolated Singularities; 6.7 On Extending Inverse Mappings to the Boundary; 6.8 On Extending Direct Mappings to the Boundary; 6.9 Consequences for Quasiextremal Distance Domains; 6.10 On Singular Null Sets for Extremal Distances; Chapter7 Strong Ring Solutions of Beltrami Equations; 7.1 Introduction
7.2 General Existence Lemma and Corollaries
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