Diffeomorphic approximation of piecewise affine homeomorphisms

Abstract

Given any $f$ a locally finitely piecewise affine homeomorphism of $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ onto $\hat{I}”\subset {\mathbb{R}}^d$ (for $d=3,4$) such that $f\in W^{1,p}(\hat{I}\text{c◯},{\mathbb{R}}^d)$ and $f^{-1}\in W^{1,q}(\hat{I}”,{\mathbb{R}}^d)$, $1\le p,q<\infty$ and any $\hat{I}\mu >0$ we construct a diffeomorphism $\tilde{f}$ such that $$\Vert f-\tilde{f}{\Vert }_{W^{1,p}(\hat{I}\text{c◯},{\mathbb{R}}^d)}+\Vert f^{-1}-{\tilde{f}}^{-1}{\Vert }_{W^{1,q}(\hat{I}”,{\mathbb{R}}^d)}<\hat{I}\mu .$$