Quasihyperbolic boundary conditions and Poincaré domains

We prove that a domain in ℝⁿ whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient β ≤ 1 is a (q, p)-Poincaré domain for all p and q satisfying p ∈ [1, ∞) ∩ (n - nβ, n) and q ∈ [p, βp*), where p* = np/(n - p) 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.