Every homeomorphism h: X -> Y between planar open sets that belongs to the Sobolev class W1;p(X;Y), 1 < p < 1, can be approximated in the Sobolev norm by Cinfinity -smooth dieomorphisms.
Iwaniec, Tadeusz; Kovalev, Leonid V.; and Onninen, Jani, "Diffeomorphic Approximation of Sobolev Homeomorphisms" (2010). Mathematics Faculty Scholarship. Paper 53.
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