On The Properties Of Several Classes Of Harmonic Mappings

Let D denote a domain in the complex plane C.A twice continuously differentiable complex-valued function F=u+iv in D is called harmonic if F satisfies the Laplacian equation:?F = 0,where ?=4(?)Obviously,harmonic mappings are a generaliza-tion of analytic functions.It is known that quasiconformal mappings are also a generalization of analytic functions.In 1968,Martio posed the concept of harmonic quasiconformal mappings in C.Since then,harmonic quasiconformal mappings have attracted much attention.The main aim of this thesis is to consider the Lipschitz continuity of the quasiconformal solutions of several classes of partial differential equations,and these solutions are generalizations of harmonic mappings.This thesis consists of six chapters,and its arrangement is as follows.In Chapter one,we provide the background of the studied problems and the statement of the obtained results.In Chapter two,we demonstrate the Lipschitz continuity of the(K,K')-quasiconformal solutions of the Poisson equation ??=g in D.In Chapter three,we get the Lipschitz,coLipschitz and biLipschitz con-tinuity of quasiconformal solutions of the non-homogeneous Yukawa equations fzz(z)=(?.(z)+?(z)fz(z))f(z)with respect to the hyperbolic metric and the quasihyperbolic metric,respectively.As an application,the corresponding area distortions of measurable sets under the quasiconformal solutions of the men-tioned Yukawa equations are also discussed.In Chapter four,we first discuss the expression and the uniqueness of the solutions to a class of inhomogeneous biharmonic Dirichlet problems,and then prove their Lipschitz continuity.In Chapter five,we obtain the Schwarz-Pick type inequality for a-harmonic functions in D and estimate their coefficients.Also,their Lipschitz continuity is investigated.In Chapter six,the main purpose is to establish the three-quarter theorem for a class of polyharmonic mappings.