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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.06026 |
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| _version_ | 1866912291312631808 |
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| author | De Masi, Luigi Gasparetto, Carlo |
| author_facet | De Masi, Luigi Gasparetto, Carlo |
| contents | We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $μ$ satisfying $\mathcal{A}μ=0$. When $\mathcal{A}$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of $μ$. Finally, we show that operators in the above class satisfy a Lusin-type property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06026 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures De Masi, Luigi Gasparetto, Carlo Analysis of PDEs We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $μ$ satisfying $\mathcal{A}μ=0$. When $\mathcal{A}$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of $μ$. Finally, we show that operators in the above class satisfy a Lusin-type property. |
| title | Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.06026 |