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On the stability of \(\varphi \)-uniform domains

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Abstract

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain \(G \subset {\mathbb R}^n\). In the sequel, we investigate a class of domains, so called \(\varphi \)-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism \(\varphi \) from \([0,\infty )\) to itself. Finally, we discuss a number of stability properties of \(\varphi \)-uniform domains. In particular, we show that the class of \(\varphi \)-uniform domains is stable in the sense that removal of a geometric sequence of points from a \(\varphi \)-uniform domain yields a \(\varphi _1\)-uniform domain.

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Acknowledgments

This research was started in the fall of 2008 when the third author was visiting the University of Turku, Finland, supported by CIMO, grant number TM-08-5606. The third author also acknowledges the support of the National Board for Higher Mathematics, DAE, India, during his post-doctoral study at IIT Madras. The work of the first author was supported by the Graduate School of Analysis and its Applications, Finland. The work of the second author was supported by the Academy of Finland grant of Matti Vuorinen Project number 2600066611 and by Hunan Provincial Innovation Foundation For Postgraduate, China. The authors thank the referees who have made valuable comments on various versions of this manuscripts.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Turku, 20014 , Turku, Finland

    R. Klén & M. Vuorinen

  2. Department of Mathematics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Y. Li

  3. Department of Mathematics, Indian Institute of Technology Indore, Indore, 452 017, India

    S. K. Sahoo

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  1. R. Klén
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  2. Y. Li
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  3. S. K. Sahoo
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  4. M. Vuorinen
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Correspondence to R. Klén.

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Communicated by A. Constantin.

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Klén, R., Li, Y., Sahoo, S.K. et al. On the stability of \(\varphi \)-uniform domains. Monatsh Math 174, 231–258 (2014). https://doi.org/10.1007/s00605-013-0576-0

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  • DOI: https://doi.org/10.1007/s00605-013-0576-0

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