Diffeomorphic Approximation of $W^{1,1}$ Planar Sobolev Homeomorphisms

Let $Ω\subseteq\mathbb R^2$ be a domain and let $f\in W^{1,1}(Ω,\mathbb R^2)$ be a homeomorphism (between $Ω$ and $f(Ω)$). Then there exists a sequence of smooth diffeomorphisms $f_k$ converging to $f$ in $W^{1,1}(Ω,\mathbb R^2)$ and uniformly.

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