Gutlyanskii, V., Ryazanov, V., Srebro, U., Yakubov, E.
2012, XIV, 302 p.
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Features a unified geometric approach based on the modulus method that is effectively applied to solving the Beltrami equation problems
Presents recent developments in the theory of Beltrami equations, especially on degenerate and alternating Beltrami equations
Discusses new concepts refining the analysis of problems related to the Beltrami equation, as well as applications of new research tools
Authors are well-known specialists in geometric function theory and elliptic differential equations
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.
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.