Diffeomorphic approximation of piecewise affine homeomorphisms

TL;DR

This paper develops a method to approximate piecewise affine homeomorphisms with smooth diffeomorphisms in Sobolev spaces, ensuring the approximation is arbitrarily close in both the function and inverse function norms.

Contribution

It introduces a technique to approximate piecewise affine homeomorphisms by diffeomorphisms in dimensions 3 and 4 within Sobolev spaces, preserving invertibility.

Findings

Approximation in Sobolev spaces for dimensions 3 and 4

Diffeomorphic approximation close in Sobolev norms

Applicable to locally finitely piecewise affine homeomorphisms

Abstract

Given any $f$ a locally finitely piecewise affine homeomorphism of $\Omega \subset \mathbb{R}^d$ onto $\Delta \subset \mathbb{R}^d$ (for $d=3, 4$) such that $f\in W^{1,p}(\Omega, \mathbb{R}^d)$ and $f^{-1}\in W^{1,q}(\Delta, \mathbb{R}^d)$, $1\leq p ,q < \infty$ and any $\epsilon >0$ we construct a diffeomorphism $\tilde{f}$ such that $$\|f-\tilde{f}\|_{W^{1,p}(\Omega,\mathbb{R}^d)} + \|f^{-1}-\tilde{f}^{-1}\|_{W^{1,q}(\Delta,\mathbb{R}^d)} < \epsilon.$$