Sobolev mappings of Euclidean space and product structure

Abstract

We consider bounded open connected sets $Ω_1, Ω_2 \subset \mathbb{R}^n$ and Sobolev maps $f: Ω_1 \times Ω_2 \subset \mathbb{R}^n \times \mathbb{R}^n$, such that for almost every $x \in Ω_1 \times Ω_2$ 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.

Figures (1)

• Figure 1: A $T_5$ configuration. The lines drawn are rank-one lines

Theorems & Definitions (49)

• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Remark 1.10
• Remark 1.11
• Theorem 1.12
• Corollary 1.14
• Lemma 2.1
• proof