Abstract
Every homeomorphism \({h : \mathbb X \to \mathbb Y}\) between planar open sets that belongs to the Sobolev class \({\fancyscript{W}^{1,p} (\mathbb X, \mathbb Y), 1 < p < \infty}\), can be approximated in the Sobolev norm by \({\fancyscript{C}^\infty}\)-smooth diffeomorphisms.
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Iwaniec, T., Kovalev, L.V. & Onninen, J. Diffeomorphic Approximation of Sobolev Homeomorphisms. Arch Rational Mech Anal 201, 1047–1067 (2011). https://doi.org/10.1007/s00205-011-0404-4
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DOI: https://doi.org/10.1007/s00205-011-0404-4