Diffeomorphic Approximation of $W^{1,1}$ Planar Sobolev Homeomorphisms

TL;DR

This paper proves that any Sobolev homeomorphism in the plane can be uniformly approximated by smooth diffeomorphisms in the $W^{1,1}$ norm, providing a bridge between Sobolev maps and smooth diffeomorphisms.

Contribution

It establishes the existence of smooth diffeomorphic approximations to $W^{1,1}$ planar homeomorphisms, extending the understanding of approximation in Sobolev spaces.

Findings

Existence of smooth diffeomorphisms approximating Sobolev homeomorphisms

Uniform convergence in addition to $W^{1,1}$ convergence

Advances the theory of Sobolev homeomorphisms in the plane

Abstract

Let $\Omega\subseteq\mathbb R^2$ be a domain and let $f\in W^{1,1}(\Omega,\mathbb R^2)$ be a homeomorphism (between $\Omega$ and $f(\Omega)$). Then there exists a sequence of smooth diffeomorphisms $f_k$ converging to $f$ in $W^{1,1}(\Omega,\mathbb R^2)$ and uniformly.