Sobolev mappings of Euclidean space and product structure

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.