Beltrami Coefficient And The Boundary Value Problems Of Quasiconformal Mappings

In this paper we mainly study the following three problems:In the second part, by studying the properties of the functions of class N, we obtain some conditions for functions that can be extended inward into class N; point out that one sufficient condition which given by Reich in his paper is not neccessary condition ; point out that inner extension is not unique; give some sufficient conditions for functions that their essential norms reduce after entending but still belong to class N.In the third part, the properties of dilatation function of Beurling-Ahlfors extension from sense-preserving homeomorphism on real axis to the upper half plane are studied. On the upper half plan and near the real axis, the dilatation functions are estimated respectively. Some latest results given by Zheng Xueliang are improved.In the forth part, we construct an analytic function g(z) which induced by given Beltrami coefficient and satisfies the inequality: ||Lu|| ≤||g|| B2 ≤3 ||Lu||.