Complex Analytic Methods for Partial Differential Equations: An Introductory Text

This is an introductory text for beginners who have a basic knowledge in complex analysis, functional analysis and partial differential equations. The Riemann and Riemann-Hilbert boundary value problems are discussed for analytic functions, for generalized Cauchy-Riemann systems and for generalized Beltrami systems. Related problems, such as the Poincare problem, pseudoparabolic systems, half Dirichlet problems for the Dirac operator in Clifford analysis and 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.