Complex Analytic Methods for Partial Differential Equations

This is an introductory text for beginners who have a basic knowledge of complex analysis, functional analysis and partial differential equations. Riemann and Riemann-Hilbert boundary value problems are discussed for analytic functions, for inhomogeneous Cauchy-Riemann systems as well as for generalized Beltrami systems. Related problems such as the Poincaré problem, pseudoparabolic systems and complex elliptic second order equations are also considered. Estimates for solutions to linear equations existence and uniqueness results are thus available for related nonlinear problems; the method is explained by constructing entire solutions to nonlinear Beltrami equations. Often problems are discussed just for the unit disc but more general domains, even of multiply connectivity, are involved.

Sample Chapter(s)
Chapter 1: Function Theoretical Tools (2,408 KB)

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• Function Theoretical Tools:
  • Cauchy Integrals
  • Green Functions and Schwarz Operators
  • Riemann Boundary Value Problem
  • Riemann-Hilbert Boundary Value Problem
• Inhomogeneous Cauchy-Riemann Systems:
  • Integral Representations
  • Properties of Integral Operators
• Boundary Value Problems for Generalized Beltrami Equations:
  • Generalized Beltrami Equation
  • Riemann Boundary Value Problem
  • Riemann-Hilbert Boundary Value Problem
  • Poincaré Boundary Value Problem
  • Discontinuous Boundary Value Problems
• Other Equations and Systems Related to the Beltrami Equation:
  • Initial Boundary Value Problem for Pseudoparabolic Equations
  • Initial Boundary Value Problem for a Composite Type System
  • Entire Solution to Nonlinear Generalized Beltrami Equations
• Higher Order Complex Differential Equations and Equations in Several Complex Variables:
  • Elliptic Second Order Equations
  • Higher Order Equations
  • First Order Systems in Two Complex Variables with Analytic Coefficients
  • The Schwarz-Poisson Formula for Polydiscs