Quasihyperbolic boundary conditions and Poincaré domains

Pekka Koskela, Jani Onninen, Jeremy T. Tyson

Research output: Contribution to journalArticle

17 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)811-830
Number of pages20
JournalMathematische Annalen
Volume323
Issue number4
DOIs
StatePublished - Dec 1 2002
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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