Quasiconformal mappings and the rank of $\tfrac {dDf}{d|Df|}$ for $f\in BV(\mathbb{R}^n; \mathbb{R}^n)$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916281316278272 |
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| author | Lahti, Panu |
| author_facet | Lahti, Panu |
| contents | We define a relaxed version $H_f^{\textrm{fine}}$ of the distortion number $H_f$ that is used to define quasiconformal mappings. Then we show that for a BV function $f\in BV(\mathbb{R}^n;\mathbb{R}^n)$, for $|Df|$-a.e. $x\in\mathbb{R}^n$ it holds that $H_{f^*}^{\textrm{fine}}(x)<\infty$ if and only if $\tfrac{dDf}{d|Df|}(x)$ has full rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05824 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasiconformal mappings and the rank of $\tfrac {dDf}{d|Df|}$ for $f\in BV(\mathbb{R}^n; \mathbb{R}^n)$ Lahti, Panu Classical Analysis and ODEs Metric Geometry We define a relaxed version $H_f^{\textrm{fine}}$ of the distortion number $H_f$ that is used to define quasiconformal mappings. Then we show that for a BV function $f\in BV(\mathbb{R}^n;\mathbb{R}^n)$, for $|Df|$-a.e. $x\in\mathbb{R}^n$ it holds that $H_{f^*}^{\textrm{fine}}(x)<\infty$ if and only if $\tfrac{dDf}{d|Df|}(x)$ has full rank. |
| title | Quasiconformal mappings and the rank of $\tfrac {dDf}{d|Df|}$ for $f\in BV(\mathbb{R}^n; \mathbb{R}^n)$ |
| topic | Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2406.05824 |