内容简介
This book consists of select works of the author, which include most important results about complex analytic theory, methods and applications obtained by the author in recent 25 years, mainly properties of solutions and various boundary value problems for nonlinear elliptic equations and systems, parabolic equations and systems, hyperbolic and mixed complex equations with parabolic degeneracy.In other words, a large portion of the works is devoted to boundary value problems for general elliptic complex equations of first and second order, initial-boundary value problems for nonlinear parabolic complex equations and systems of second order including some equations and systems in higher dimensional domains, and properties of solutions for hyperbolic complex equations of second order. Moreover, some results about second order complex equations of mixed (elliptic-hyperbolic) type are introduced. Applications of nonlinear complex analysis to continuum mechanics, and approximate methods of elliptic complex equations axe also investigated.
目录
Preface
Chapter 1 Foundational Theorems of Nonlinear Quasiconformal Mappings and Quasiconformal Shift Theorems
1.1 Existence Theorems of Nonlinear Quasiconformal Mappings in Multiply Connected Domains
1.2 Uniqueness Theorems of Nonlinear Quasiconformal Mappings in Multiply Connected Domains
1.3 General Quasiconformal Shift Theorems in Multiply Connected Domains
1.4 Quasiconformal Shift Theorems with Other Shift Conditions
Chapter 2 Boundary Value PrOblems for Nonlinear Elliptic Complex Equations and Systems
2.1 Reduction of General Uniformly Elliptic Systems of First Order Equations to Standard Complex Form
2.2 The Well-Posedness of Riemann-Hilbert Problem with Nonsmooth Boundary
2.3 A Prior Estimate of Solutions for Problems B and B'
2.4 Uniqueness of Solutions and Solvability for Problems B and B'
2.5 Formulation of Oblique Derivative Problems of Second Order Systems and Statement of Main Theorem
2.6 Formulation of Modified Problem of First Order System and Integral expression of Its Solutions
2.7 Estimates of Solutions for Modified Boundary Value Problem of First Order System
2.8 Solvability of Modified Problem of First Order System and Oblique Derivative Problem of Second Order System
Chapter 3 Discontinuous Boundary Value Problems for Analytic Functions and Nonlinear Elliptic Equations
3.1 General Discontinuous Boundary Value Problem for Analytic Functions in Upper Half-Plane
3.2 General Discontinuous Boundary Value Problem for Analytic Functions in Unit-Disk
3.3 Formulation of Discontinuous Irregular Oblique Derivative Problems for Nonlinear Elliptic Equations
3.4 Uniqueness and Estimates of Solutions of General Discontinuous Oblique Derivative Problems
3.5 Solvability of General Discontinuous Oblique Derivative Problems
3.6 General Continuous Oblique Derivative Problems
Chapter 4 Approximate Methods for Solving Elliptic Systems and Their Error Estimate
4.1 Newton Imbedding Method of Riemann-Hilbert Problem for Nonlinear Elliptic Systems of First Order
4.2 Error Estimates of Approximate Solutions of Riemann-Hilbert Problem for Elliptic Systems of First Order
4.3 Transformation of Elliptic Systems and Compound Boundary Value Problem
4.4 Variation-Difference Method of Solving Compound Boundary Value Problem
4.5 Variation-Difference Method of Oblique Derivative Problem for Second Order Elliptic Equations
Chapter 5 Oblique Derivative Problems for Degenerate Equations of Mixed Type in Multiply Connected Domains
5.1 Formulation of Oblique Derivative Problem for Degenerate Equations of Mixed Type
5.2 Representation of Solutions of Oblique Derivative Problem for Degenerate Equations of Mixed Type
5.3 Uniqueness of Solutions of Oblique Derivative Problem for Degenerate Equations of Mixed Type
5.4 Solvability of Oblique De
