The Character Of John Disks

John domain was led into by F.John while studying elastic mechanics in 1961, it can be looked upon as one sided quasidisk, intuitively, the John condition prohibits outward pointing cusps, and contain the unique geometry characteristic, it already be a study object in complex anallytic dynamical systems, approach theory and elastic mechanics, while the points of John domain's boundary was more than one, it was gived out definition as John disk by R.Nakki and J.Vaisal in 1991.In this thesis, we obtain various decision methods of John disks.The thesis is divided into three chapters.In chapter 1, we introduce the development of the John disks, then we give out mathematics signs in common use, a sufficient condition of quasidisk, two sufficient conditions of John disk, and definitons of bounded and unbounded John disk,linearly connected domains.In chapter 2, we make use of Schwarzian derivative to give out a decision theory of quasidisk, then we make use of pre-Schwarzian derivative to give out a decision method of John disks.In chapter 3, we introduce a functionalσ_ζ(r),by using a comparison theorem of second order differential equations'solutions, we obtain a sufficient condition of John disks and resolve a problem which is proposed by Hag.