We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space W1,2 and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.
Iwaniec, Tadeusz; Kovalev, Leonid V.; and Onninen, Jani, "Hopf Differentials and Smoothing Sobolev Homeomorphisms" (2010). Mathematics Faculty Scholarship. 64.
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