Boundary Value Problem

This monograph mainly deals with several boundary value problems for linear and nonlinear elliptic equations and systems by using function theoretic methods. The established theory is systematic, the considered equations and systems, boundary conditions and domains are rather general. Various methods are used. As an application, the existence of nonlinear quasiconformal mappings onto canonical domains is proved. Graduate students with basic knowledge in partial differential equations and complex analysis and specialists in partial differential equations or complex analysis should find this book of use.

Table of Contents:
Part 1 Boundary value problems for simple complex equations: some boundary value problems for harmonic functions; some boundary value problems for simple first order complex equations; some boundary value problems for simple second order equations. Part 2 Boundary value problems for complex elliptic equatioms of first order: reductions of elliptic systems of two first order equations; properties of solutions for the elliptic complex equation of first order; the Haseman boundary value problem for elliptic complex equations of first order; the Riemann-Hibert problem for the elliptic complex equation of first order; applications of boundary value problems to nonlinear quasiconformal mappings; the compound boundary value problem for elliptic complex equations. Part 3 Boundary value problems for elliptic equations of second order: reduction of elliptic equations of second order; extremum principles for elliptic equations of second order; representation theorem and compactness principle of solutions for equations of second order; the regular oblique derivative problem for elliptic equations of second order; the irregular oblique derivative problems for elliptic equations of second order; the Poincare boundary value problem for elliptic equations of second order. Part 4 Boundary value problem with piecewise continuous coefficients for elliptic equations and systems: boundary value problems with piecewise continuous coefficients for analytic functions; the Riemann-Hilbert problem with piecewise continuous coefficients for complex equations of first order; the Riemann boundary value problem with piecewise continuous coefficients for complex equations on closed curves; the Poincare problem with piecewise continuous coefficients for elliptic equations of second order. Part 5 Boundary value problems for elliptic systems of two second order equations: reduction and conditions for elliptic systems of two second order equations; the Dirichlet problem and the Neumann problem for elliptic complex equations of second order; the oblique derivative problem for elliptic complex equations of second order; the Riemann-Hilbert problem for complex equations of second order; boundary value problems for another class of second order complex equations. Part 6 Boundary value problems for elliptic systems of several equations: reductions of elliptic systems of several equations fo first and second order; boundary value problem for nonlinear complex systems of first order; boundary value problems for complex systems of several linear equations of first and second order; the oblique derivative problem for complex systems of nonlinear second order equations.