Smooth homeomorphic approximation of piecewise affine homeomorphisms

Given any $f$ a locally finitely piecewise affine homeomorphism of $Ω\subset \rn$ onto $Δ\subset \rn$ in $W^{1,p}$, $1\leq p < \infty$ and any $ε>0$ we construct a smooth injective map $\tilde{f}$ such that $\|f-\tilde{f}\|_{W^{1,p}(Ω,\rn)} < ε$.

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