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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2506.13718 |
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| _version_ | 1866914171643232256 |
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| author | Takáč, Jakub |
| author_facet | Takáč, Jakub |
| contents | We show that Lang's Flat Chain Conjecture (that is, without requiring finite mass of the underlying currents) fails for metric $k$-currents in $\mathbb{R}^d$ whenever $d\geq 2$ and $k\in\{1, \dots, d\}$. In all other cases, it holds. The original conjecture due to Ambrosio and Kirchheim remains open. We first connect Lang's conjecture to a regularity statement concerning the prescribed Jacobian equation near $L^\infty$. We then show that the equation does not have the required regularity. For a Lipschitz vector field $π$, its derivative $\mathrm{D}π$ exists a.e. and is identified with a matrix. Our non-regularity results for the prescribed Jacobian equation quantify how "small" the set
\begin{equation*}
\operatorname{conv}(\{\operatorname{det}\mathrm{D} π: \operatorname{Lip}(π)\leq L\})\subset L^\infty
\end{equation*}
is for every $L>0$. The symbol "$\operatorname{conv}$" stands for the convex hull. The "smallness" is quantified in topological terms and is used to show that Lang's Flat Chain Conjecture fails. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13718 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Failure of Lang's Flat Chain Conjecture and non-regularity of the prescribed Jacobian equation Takáč, Jakub Functional Analysis Analysis of PDEs 49Q15 (Primary) 26B10, 35B65 (Secondary) We show that Lang's Flat Chain Conjecture (that is, without requiring finite mass of the underlying currents) fails for metric $k$-currents in $\mathbb{R}^d$ whenever $d\geq 2$ and $k\in\{1, \dots, d\}$. In all other cases, it holds. The original conjecture due to Ambrosio and Kirchheim remains open. We first connect Lang's conjecture to a regularity statement concerning the prescribed Jacobian equation near $L^\infty$. We then show that the equation does not have the required regularity. For a Lipschitz vector field $π$, its derivative $\mathrm{D}π$ exists a.e. and is identified with a matrix. Our non-regularity results for the prescribed Jacobian equation quantify how "small" the set \begin{equation*} \operatorname{conv}(\{\operatorname{det}\mathrm{D} π: \operatorname{Lip}(π)\leq L\})\subset L^\infty \end{equation*} is for every $L>0$. The symbol "$\operatorname{conv}$" stands for the convex hull. The "smallness" is quantified in topological terms and is used to show that Lang's Flat Chain Conjecture fails. |
| title | Failure of Lang's Flat Chain Conjecture and non-regularity of the prescribed Jacobian equation |
| topic | Functional Analysis Analysis of PDEs 49Q15 (Primary) 26B10, 35B65 (Secondary) |
| url | https://arxiv.org/abs/2506.13718 |