Properties of John disks.

Abstract

A domain D in euclidean n-space $R\sp n$ is said to be a John Domain if there exist a point $x\sb0$ in D and a constant $c\geq1$ such that point $x\sb1$ in D can be joined in D to $x\sb0$ by an arc $\gamma$ such that the length of the subarc from $x\sb1$ to x in $\gamma$ is bounded above by c times the distance from x to the boundary $\partial D$ of D. This class was first studied by Fritz John in 1961 in connection with his work on elasticity and local quasi-isometries. In this thesis we study John disks, the John domains D in $R\sp2$ for which $\partial D$ is a Jordan curve, and we give new conformally invariant, geometric and function theoretic properties and characterizations for this class. The conformally invariant properties involve harmonic and hyperbolic measure, the geometric conditions concern quasidisks and a quasiextremal distance property, and the function theoretic properties consist of an analogue of the Bernstein inequality and the conformal mapping of the exterior of the unit disk onto the exterior of D. In the final chapter we characterize the compact sets on which the above analogue of the Bernstein inequality holds in terms of the exterior mapping function.