Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms

TL;DR

This paper demonstrates that planar bi-Lipschitz orientation-preserving homeomorphisms can be approximated in the Sobolev space by smoother or piecewise affine homeomorphisms, preserving orientation.

Contribution

It introduces a method to approximate bi-Lipschitz homeomorphisms with smooth or piecewise affine ones in the $W^{1,p}$ norm, including their inverses.

Findings

Approximation in $W^{1,p}$ norm achieved for bi-Lipschitz homeomorphisms.

Preservation of orientation during approximation.

Applicable to both smooth and piecewise affine approximations.

Abstract

We show that a planar bi-Lipschitz orientation-preserving homeomorphism can be approximated in the $W^{1,p}$ norm, together with its inverse, with an orientation-preserving homeomorphism which is piecewise affine or smooth.