Lowner微分方程与调和拟共形映射

Lowner Different Equation and Harmonic Quasiconformal Mapping

文章研究Lowner微分方程dw dt=F(w,t)中F(ω,t)与方程解ω=f(z,t)之间的关系问题.利用F(ω,t)的可微性,得到具有调和同胚解的Lowner微分方程所需的必要条件,并证明当方程存在形如ω=f(z,t)=u(x,y,t)+iy的K(t)-调和拟共形映射解时,F(ω,t)是调和映射.特别地,若u(z,t)∈C2(H,[0,T]),则F(ω,t)是调和拟共形形变.

This paper studies the relationship between F(ω,t)and equation solutionsω=f(z,t)in the Lown⁃er differention equation.Using the differentiability of F(ω,t),the necessary conditions for the Lowner differen⁃tial equation with the solution of the harmonic homeomorphism are obtained,and proves that the function F(ω,t)is harmonic when the equation has a the solution of the K(t)-harmonic quasiconformal mapping of the formω=f(z,t)=u(x,y,t)+iy.In particular,if u(z,t)∈C2(H,[0,T]),then F(ω,t)is harmonic quasiconfor⁃mal deformation.