A decomposition for Borel measures $μ\le \mathcal{H}^{s}$
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866913676819169280 |
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| author | Detaille, Antoine Ponce, Augusto C. |
| author_facet | Detaille, Antoine Ponce, Augusto C. |
| contents | We prove that every finite Borel measure $μ$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $μ\lfloor_{E_k}$ that are bounded from above by the Hausdorff content $\mathcal{H}_\infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to show the existence of solutions of a Dirichlet problem involving an exponential nonlinearity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_15902 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A decomposition for Borel measures $μ\le \mathcal{H}^{s}$ Detaille, Antoine Ponce, Augusto C. Classical Analysis and ODEs Primary: 28A78, Secondary: 28A12 We prove that every finite Borel measure $μ$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $μ\lfloor_{E_k}$ that are bounded from above by the Hausdorff content $\mathcal{H}_\infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to show the existence of solutions of a Dirichlet problem involving an exponential nonlinearity. |
| title | A decomposition for Borel measures $μ\le \mathcal{H}^{s}$ |
| topic | Classical Analysis and ODEs Primary: 28A78, Secondary: 28A12 |
| url | https://arxiv.org/abs/2010.15902 |