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The Beltrami Equation

Part of the Developments in Mathematics Book Series (volume 26)

DOI: 10.1007/978-1-4614-3191-6

ISBN: 978-1-4614-3190-9

Publisher: Springer New York - Springer Journals

Mar 12, 2012

1 - 10 of 10 Chapters

LitStream

Chapter 1Introduction

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 1 –4

[Let ℂ be the complex plane. In the complex notation w = u+iv and z = x+iy, the Beltrami equation in a domain D ⊂ ℂ has the form B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...
LitStream

Chapter 2Preliminaries

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 5 –45

[The class BMO was introduced by John and Nirenberg in the paper [122] and soon became an important concept in harmonic analysis, partial differential equations, and related areas; see, e.g., [21, 24, 84, 103, 200] and [239].]
LitStream

Chapter 3The Classical Beltrami Equation ||µ||∞<1

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 47 –53

[Consider the Beltrami equation (B) in Sect. 1.1 with ||µ||∞<1, and let f : D → ℂ be its homeomorphic solution. Since |µ| < 1 a.e., f is sense preserving, and since ||µ||∞<1, f is a quasiconformal mapping.]
LitStream

Chapter 4The Degenerate Case

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 55 –76

[Recall that in our notation, a µ-homeomorphism in a domain D,D ⊂ ℂ is an ACL homeomorphic solution of (B) in D; see Sect. 1.5. For some functions µ with |µ(z)| ≤ 1 a.e. and ||µ||∞ = 1, there are no µ-homeomorphisms, i.e., homeomorphic solutions of (B), as illustrated below in Sects. 4.1.1 and...
LitStream

Chapter 5BMO- and FMO-Quasiconformal Mappings

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 77 –95

[In this chapter, the BMO-quasiconformal and BMO-quasiregular mappings in the plane are studied. This includes distortion, existence, uniqueness, representation, integrability, convergence and removability theorems, the reflection principle, boundary behavior, and mapping properties.]
LitStream

Chapter 6Ring Q-Homeomorphisms at Boundary Points

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 97 –128

[In this chapter, we introduce and study plane ring Q-homeomorphisms. This study is then applied to deriving general principles on the existence of strong ring solutions to the Beltrami equation extending and strengthening earlier results; see the next chapter.]
LitStream

Chapter 7Strong Ring Solutions of Beltrami Equations

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 129 –151

[In this chapter we give a series of criteria for the existence of strong ring solutions, in particular, in terms of finite mean oscillation majorants for tangential dilatations. Moreover, we derive an extension of the well-known Lehto existence theorem and show that the latter implies the main...
LitStream

Chapter 8On the Dirichlet Problem for Beltrami Equations

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 153 –169

[Boundary problems for the Beltrami equation (B) are due to the famous dissertation of Riemann who considered a particular case of analytic functions when (µz) ≡ 0 and to the works of Hilbert (1904, 1924) who studied the corresponding system of Cauchy–Riemann for the real and imaginary parts of...
LitStream

Chapter 9On the Beltrami Equations with Two Characteristics

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 171 –182

[The Beltrami equations of the first type 9.1.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...
LitStream

Chapter 10Alternating Beltrami Equation

Gutlyanskii, Vladimir; Ryazanov, Vladimir; Srebro, Uri; Yakubov, Eduard

Pages: 183 –208

[Roughly speaking, this term refers to equation (B) in the case where μ is a measurable complex-valued function in a domain D in ℂ with |μ| < 1 a.e. in parts of D and |μ| > 1 a.e. in other parts of D. With further assumptions on μ, every nonconstant solution f : D → ℂ of (B), i.e., an ACL mapping...

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