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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.05979 |
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| _version_ | 1866910043472920576 |
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| author | Kleiner, Bruce Müller, Stefan Székelyhidi Jr., László Xie, Xiangdong |
| author_facet | Kleiner, Bruce Müller, Stefan Székelyhidi Jr., László Xie, Xiangdong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05979 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sobolev mappings of Euclidean space and product structure Kleiner, Bruce Müller, Stefan Székelyhidi Jr., László Xie, Xiangdong Analysis of PDEs Differential Geometry 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. |
| title | Sobolev mappings of Euclidean space and product structure |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2603.05979 |