Quasihyperbolic boundary conditions and Poincaré domains

We prove that a domain in whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient is a (q,p)-\Poincare domain for all p and q satisfying and , where denotes the Sobolev conjugate exponent. An elementary example shows that the given ranges for p and q are sharp. The proof makes use of estimates for a variational capacity. When p=2 we give an application to the solvability of the Neumann problem on domains with irregular boundaries. We also discuss the relationship between this growth condition on the quasihyperbolic metric and the s-John condition. Received: 2 May 2000 / Published online: 17 June 2002