The Solutions And Properties Of The Lowner Differential Equation And (?)-Poisson Equation

Harmonic mapping and quasiconformal mapping are two important branches in the theory of monomorphic functions.This paper mainly studies the problems related to the Lowner differential equation in the theory of quasiconformal mapping,and the specific forms and properties of the solution of (?)-Poisson equation.The thesis is divided into four chapters:The first chapter mainly introduces the background knowledge of plane harmonic mapping,quasiconformal mapping and harmonic quasiconformal mapping,as well as the relevant concepts and historical background of the Lowner differential equation and the (?)-Poisson equation.In the second chapter,we mainly study the problems related to the solution of the Lowner differential equation.In particular,the necessary conditions of the Lowner differential equation with the solution of the harmonic homeomorphism are discussed.Further,we have proved that if the Lowner differential equation has the solution ?=f(z,t)=u(x,y,t)+iy,which is k(t)-harmonic quasiconformal mapping,then the vector field F(?,t) must be harmonic.In the third chapter,we mainly discuss the solution of the (?)-Poission equation,namely (?)-harmonic mapping.We give the necessary condition of the radial function ?=g(s)e^it? C² on the ring domain,which is (?)-harmonic mapping,then we give its specific form.In addition,we also give the specific form of second-order differential operator D(?,z)of this class of functions,where D(?,z) is a class of operators related to the (?)-Poisson equation.The fourth chapter summarizes this paper.