Sobolev mappings of Euclidean space and product structure

TL;DR

This paper proves that Sobolev maps with invertible differentials on product domains in Euclidean space are split into functions of individual variables for dimensions n≥2, but not for n=1, extending previous results and exploring approximate splitting.

Contribution

It establishes conditions under which Sobolev maps on product domains are necessarily split, highlighting differences between dimensions and Sobolev spaces, and discusses approximate splitting.

Findings

For n≥2, Sobolev maps with invertible differentials are split functions.

The splitting conclusion fails for n=1, even with bi-Lipschitz and area-preserving assumptions.

Approximate splitting results are discussed for sequences of maps approaching split maps.

Abstract

We consider bounded open connected sets $\Omega_1, \Omega_2 \subset \mathbb{R}^n$ and Sobolev maps $f: \Omega_1 \times \Omega_2 \subset \mathbb{R}^n \times \mathbb{R}^n$, such that for almost every $x \in \Omega_1 \times \Omega_2$ the weak differential $\nabla f(x)$ is invertible and preserves or swaps the spaces $\mathbb{R}^n \times \{0\}$ and $\{0\} \times \mathbb{R}^n$. We show that if $n \ge 2$ and $f \in W^{1,2}$ then $f$ is split, i.e., $f(x_1, x_2) = (f_1(x_1), f_2(x_2))$ or $f(x_1, x_2) = (f_2(x_2), f_1(x_1))$. We also show that this conclusion fails in general for $n=1$, even if we assume in addition that $f$ is bi-Lipschitz and area preserving. These results complement our previous work https://arxiv.org/abs/2403.20265, where we showed that the conclusion fails for $n \ge 2$ if the Sobolev space $W^{1,2}$ is replaced by $W^{1,p}$ for any $p < 2$. We also discuss results for approximately split maps, i.e. for sequences of maps $f_k$ such that $\nabla f_k$ approaches the set of linear invertible split maps in suitable $L^p$ spaces. This work is partly motivated by the question whether Sobolev maps defined on products of Carnot groups are split.