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

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.