The closure of planar diffeomorphisms in Sobolev spaces

TL;DR

This paper characterizes the closure properties of planar diffeomorphisms in Sobolev spaces, showing their equivalence under weak and strong convergence, and provides conditions for approximation by diffeomorphisms.

Contribution

It offers a complete description of the Sobolev closure of planar diffeomorphisms and introduces new criteria for their approximation based on connectedness of pre-images.

Findings

Weak and strong closures of planar diffeomorphisms coincide.

Sufficient conditions for approximation involve connectedness of pre-images.

The Sobolev closure is strictly contained in the INV class.

Abstract

We characterize the (sequentially) weak and strong closure of planar diffeomorphisms in the Sobolev topology and we show that they always coincide. We also provide some sufficient condition for a planar map to be approximable by diffeomorphisms in terms of the connectedness of its counter-images, in the spirit of Young's characterisation of monotone functions. We finally show that the closure of diffeomorphisms in the Sobolev topology is strictly contained in the class INV introduced by Muller and Spector.