Approximation of planar Sobolev $W^{2,1}$ homeomorphisms by Piecewise Quadratic Homeomorphisms and Diffeomorphisms

TL;DR

The paper develops a method to approximate planar Sobolev $W^{2,1}$ homeomorphisms with piecewise quadratic maps and further approximates these by diffeomorphisms, ensuring close $W^{2,1}$ norm convergence.

Contribution

It introduces a novel approximation technique for Sobolev homeomorphisms using piecewise quadratic maps and diffeomorphisms in the plane.

Findings

Piecewise quadratic homeomorphisms can approximate $W^{2,1}$ homeomorphisms up to small measure sets.

These quadratic maps can be further approximated by diffeomorphisms in the $W^{2,1}$ norm.

The approximation holds with arbitrary precision outside a set of measure $ ext{epsilon}$.

Abstract

Given a Sobolev homeomorphism $f\in W^{2,1}$ in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set of $\epsilon$ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in the $W^{2,1}$ norm on this set.