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Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains

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In the first part of this article we give intrinsic characterizations of the classes of Lipschitz and C1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C1. In the second part of the article, we study the invariance of various classes of domains of locally finite perimeter under bi-Lipschitz and C1 diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chord-arc domains with vanishing constant, in the literature) is stable under C1 diffeomorphisms. A number of other applications are also presented.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO

    Steve Hofmann & Marius Mitrea

  2. Mathematics Department, University of North Carolina, 27599, Chapel Hill, NC

    Michael Taylor

Authors
  1. Steve Hofmann
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  2. Marius Mitrea
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  3. Michael Taylor
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Corresponding author

Correspondence to Steve Hofmann.

Additional information

Communicated by Steven G. Krantz

Acknowledgements and Notes. The work of the authors was supported in part by NSF grants DMS-0245401, DMS-0653180, DMS-FRG0456306, and DMS-0456861.

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Hofmann, S., Mitrea, M. & Taylor, M. Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains. J Geom Anal 17, 593–647 (2007). https://doi.org/10.1007/BF02937431

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