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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.18819 |
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| _version_ | 1866918397998006272 |
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| author | Bianchini, Stefano Talamini, Luca |
| author_facet | Bianchini, Stefano Talamini, Luca |
| contents | Consider a piecewise affine Lipschitz map $ϕ: Ω\to \mathbb R$, where $Ω\subset \mathbb R^d$ is an open set, and assume that $x \mapsto x + t \nabla ϕ(x)$ is injective for almost every $t > 0$. In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., \textbf{7}(4), 2025) the authors conjecture that every such $ϕ$ must be locally convex. We prove the result assuming additionally $\nabla ϕ\in BV_{loc}(Ω)$, for a more general class of measure preserving maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18819 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Measure preserving maps with bounded total variation Bianchini, Stefano Talamini, Luca Analysis of PDEs Functional Analysis Consider a piecewise affine Lipschitz map $ϕ: Ω\to \mathbb R$, where $Ω\subset \mathbb R^d$ is an open set, and assume that $x \mapsto x + t \nabla ϕ(x)$ is injective for almost every $t > 0$. In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., \textbf{7}(4), 2025) the authors conjecture that every such $ϕ$ must be locally convex. We prove the result assuming additionally $\nabla ϕ\in BV_{loc}(Ω)$, for a more general class of measure preserving maps. |
| title | Measure preserving maps with bounded total variation |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2603.18819 |