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Let and be -connected Jordan domains, , with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism admits a Sobolev homeomorphic extension in . If instead has -hyperbolic growth with , we show the existence of such an extension in the Sobolev class for . Our examples show that the assumptions of rectifiable boundary and hyperbolic growth cannot be relaxed. We also consider the existence of -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. 4065–4089DOI 10.4171/JEMS/1099