On The Properties Of John Domains And Uniform Domains

In order to investigate the elasticity and injectivity theory, in1961, John introduced a class of domains which was named as John domains by Martio and Sarvas in1978. In1978, Martio and Sarvas introduced the so called uniform domains. It is well known that the properties of John domains and uniform domains in Rn have been widely studied and have many applications. The significance of these classes of domains is not restricted to the case of finite dimensional spaces as pointed out by Vaisala. We shall see that many properties of John and uniform domains also hold in general Banach spaces. As a number of tools and methods, which are useful in Rn, are not available in Banach spaces, we should find new tools and methods. The main tools come from the theory of metric spaces:the quasihyperbolic metric, the distance ratio metric and the norm metric are the basic metric structures we use to study these domains. Also curves and nearly length-minimizing curves are extensively used.The main aim of this thesis is to get certain properties of John domains, un-form domains and to study Vaisala’s theory of mappings between two domains in Banach spaces. Here classes of maps such as bilipschitz, quasiconformal and quasisymmetric maps as well as classes of domains such as uniform and John domains are some of the key notions. This thesis consists of seven chapters and the arrangement is as follows.In Chapter one, we provide the background on our research and the state-ment of our main results.In Chapter two, we investigate the union of John domains and uniform domains, respectively, in Banach spaces. Firstly, we get, under some condition, the union of two John domains is still a John domain, which generalize the corresponding result of Vaisala published in Proc. Amer. Math. Soc. in1999. Secondly, we consider the same problem for uniform domains. At last, we give the definition of the decomposition of John domains and prove that a domain is a John domain if and only if it has this decomposition property.In Chapter three, we study the stability of John domains in Banach spaces. We mainly show that the class of John domains is stable in the sense that removing a certain type of countable points from each of these domains yields a new domain which is also a John domain. As an application, we show the stability of the inner uniform domains. Finally, we get an analogue of φ-John domains.In Chapter four, we study the relation between neargeodesics and cone arcs in John domains and prove that if D E is a John domain which is homeomorphic to an inner uniform domain via a CQH mapping, then each neargeodesic in D is a cone arc.In Chapter five, we discuss some related properties of freely quasiconfor-mal mappings in Banach spaces. We consider the class of all φ-FQC (freely φ-quasiconfornial) mappings from a uniform domain D onto another uniform domain D’with bilipschitz boundary values. We show that all mappings in this class are η-quasisymmetric. As applications, we show that (1) all mappings in this class satisfy a two sided Holder condition;(2) if f is a φ-FQC mapping which maps D onto itself with identity boundary values, then there is a constant C, depending only on the function φ. such that for all χ∈D, the quasihyper-bolic distance between χ and its image f(χ) satisfies kD(χ, f(χ))≤C. Finally, replacing the class φ-FQC mappings by the class of M-QH mappings, we show that each QH mapping with bilipschitz boundary values is bilipschitz.In Chapter six, we study the relation between Apollonian inner metric and inner uniform domains. We characterize inner uniform domains in Rn in terms of Apollonian inner metric and the metric j’D when D are Apollonian. As an application, a new characterization for A-uniform domains is obtained.In Chapter seven, we investigate a class of so called φ-uniform domains, and discuss the stability property of φ-uniform domains. In particular, we show that the class of φ-uniform domains is stable in the sense that the removal of a geometric sequence of points from aφ-uniform domain yields a new φ1-uniform domain.