Two Approximation Results for Divergence Free Measures
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866916132314677248 |
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| author | Goodman, Jesse Hernandez, Felipe Spector, Daniel |
| author_facet | Goodman, Jesse Hernandez, Felipe Spector, Daniel |
| contents | In this paper we prove two approximation results for divergence free measures. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given $F \in M_b(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{div} F=0$ in the sense of distributions, there exist oriented $C^1$ loops $Γ_{i,l}$ with associated measures $μ_{Γ_{i,l}}$ such that \[ F= \lim_{l \to \infty} \frac{\|F\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)}}{n_l \cdot l} \sum_{i=1}^{n_l} μ_{Γ_{i,l}} \] weakly-star in the sense of measures and \[ \lim_{l \to \infty} \frac{1}{n_l \cdot l} \sum_{i=1}^{n_l} \|μ_{Γ_{i,l}}\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)} = 1. \] The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in \[ \left\{ F \in M_b(\mathbb{R}^d;\mathbb{R}^d): \operatorname*{div}F=0 \right\} \] with respect to the strict topology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_14079 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Two Approximation Results for Divergence Free Measures Goodman, Jesse Hernandez, Felipe Spector, Daniel Analysis of PDEs Functional Analysis In this paper we prove two approximation results for divergence free measures. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given $F \in M_b(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{div} F=0$ in the sense of distributions, there exist oriented $C^1$ loops $Γ_{i,l}$ with associated measures $μ_{Γ_{i,l}}$ such that \[ F= \lim_{l \to \infty} \frac{\|F\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)}}{n_l \cdot l} \sum_{i=1}^{n_l} μ_{Γ_{i,l}} \] weakly-star in the sense of measures and \[ \lim_{l \to \infty} \frac{1}{n_l \cdot l} \sum_{i=1}^{n_l} \|μ_{Γ_{i,l}}\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)} = 1. \] The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in \[ \left\{ F \in M_b(\mathbb{R}^d;\mathbb{R}^d): \operatorname*{div}F=0 \right\} \] with respect to the strict topology. |
| title | Two Approximation Results for Divergence Free Measures |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2010.14079 |