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To the theory of mappings of the Sobolev class with the critical index

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Abstract

It is established that any homeomorphism f of the Sobolev class \( {W}_{\mathrm{loc}}^{1,1} \) with outer dilatation \( {K}_O\left(x,f\right)\in {L}_{\mathrm{loc}}^{n-1} \) is the so-called lower Q-homeomorphism with Q(x) = KO(x, f) and also a ring Q-homeomorphism with \( Q(x)={K}_O^{n-1}\left(x,f\right) \). This allows us to apply the theory of boundary behavior of ring and lower Q-homeomorphisms. In particular, we have found the conditions imposed on the outer dilatation KO(x, f) and the boundaries of domains under which any homeomorphism of the Sobolev class \( {W}_{\mathrm{loc}}^{1,1} \) admits continuous or homeomorphic extensions to the boundary.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine

    Elena S. Afanas’eva & Vladimir I. Ryazanov

  2. Institute of Mathematics of the NAS of Ukraine, Kiev, Ukraine

    Ruslan R. Salimov

Authors
  1. Elena S. Afanas’eva
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  2. Vladimir I. Ryazanov
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  3. Ruslan R. Salimov
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Correspondence to Elena S. Afanas’eva.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 15, No. 2, pp. 154–176, January–March, 2018.

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Afanas’eva, E.S., Ryazanov, V.I. & Salimov, R.R. To the theory of mappings of the Sobolev class with the critical index. J Math Sci 239, 1–16 (2019). https://doi.org/10.1007/s10958-019-04283-0

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