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.

36 pages. Companion paper to arXiv:2403.20265