Smooth homeomorphic approximation of piecewise affine homeomorphisms

TL;DR

This paper presents a method to approximate any piecewise affine homeomorphism with a smooth injective map in the Sobolev space, ensuring the approximation is arbitrarily close in the $W^{1,p}$ norm.

Contribution

It introduces a technique to smoothly approximate piecewise affine homeomorphisms while preserving injectivity and controlling the approximation error.

Findings

Successfully approximates piecewise affine homeomorphisms with smooth injective maps

Ensures the approximation error can be made arbitrarily small in the Sobolev norm

Maintains the homeomorphic property in the approximation process

Abstract

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