On The Properties Of John Domains

In order to investigate the properties of approximation and injectivity of bi-Lipschitz mappings,in 1961,John introduced a class of domains in C which were named as John domains by Martio and Sarvgas in 1978.Since then John domains have become a hot research object.John domains play an important role in the theory of partial differential equation,quasiconformal mapping,potential theory,geometric measure theory and Fuchs group.The main aim of this paper is to establish nine equivalent conditions for a bounded domain,which is quasiconformally equivalent to a bounded John disk,to be a John disk.This result is a generalization of the main result in[1].It consists of four chapters,and the concrete arrangement is as follows.In Chapter one,we provide the background on the studied problems and the statement of our main results,especially,the definition and properties of the John domains.In Chapter two,we introduce some necessary preliminaries,including the concepts of quasihyperbolie metric,the conformal modulus of a curve family,and some related properties of them.In Chapter three,we discuss some properties of John disk under the qusasi-conformal mappings and quasisymmetric mappings.These properties will play an important role in the proof of the main results of this paper.In Chapter four,we prove nine equivalent conditions of John disk.