JournalsjemsVol. 23, No. 12pp. 4065–4089

Sobolev homeomorphic extensions

  • Aleksis Koski

    University of Jyväskylä, Finland
  • Jani Onninen

    Syracuse University, USA
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Let X\mathbb{X} and Y\mathbb{Y} be \ell-connected Jordan domains, N\ell \in \mathbb{N}, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism φ ⁣:X ⁣ ⁣onto ⁣ ⁣Y\varphi \colon \partial \mathbb{X} \xrightarrow[]{{}_{\!\!\textnormal{onto\,\,}\!\!}} \partial \mathbb{Y} admits a Sobolev homeomorphic extension h ⁣:X ⁣ ⁣onto ⁣ ⁣Yh \colon \overline{\mathbb{X}} \xrightarrow[]{{}_{\!\!\textnormal{onto\,\,}\!\!}} \overline{\mathbb{Y}} in W1,1(X,C)\mathscr{W}^{1,1} (\mathbb{X}, \mathbb{C}). If instead X\mathbb{X} has ss-hyperbolic growth with s>p1s>p-1, we show the existence of such an extension in the Sobolev class W1,p(X,C)\mathscr{W}^{1,p} (\mathbb{X}, \mathbb{C}) for p(1,2)p\in (1,2). Our examples show that the assumptions of rectifiable boundary and hyperbolic growth cannot be relaxed. We also consider the existence of W1,2\mathscr{W}^{1,2}-homeomorphic extensions with given boundary data.

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Aleksis Koski, Jani Onninen, Sobolev homeomorphic extensions. J. Eur. Math. Soc. 23 (2021), no. 12, pp. 4065–4089

DOI 10.4171/JEMS/1099