Porosity of the Free Boundary in a Class of Higher-Dimensional Elliptic Problems
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910345699786752 |
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| author | Lyaghfouri, Abdeslem |
| author_facet | Lyaghfouri, Abdeslem |
| contents | We investigate a class of n-dimensional free boundary elliptic problems which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its Hausdorff dimension being less than $n$, which in turn leads to its Lebesgue measure being zero. Our proof relies on the comparison of the solution with an appropriately constructed barrier function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17676 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Porosity of the Free Boundary in a Class of Higher-Dimensional Elliptic Problems Lyaghfouri, Abdeslem Analysis of PDEs 35J15, 35R35 We investigate a class of n-dimensional free boundary elliptic problems which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its Hausdorff dimension being less than $n$, which in turn leads to its Lebesgue measure being zero. Our proof relies on the comparison of the solution with an appropriately constructed barrier function. |
| title | Porosity of the Free Boundary in a Class of Higher-Dimensional Elliptic Problems |
| topic | Analysis of PDEs 35J15, 35R35 |
| url | https://arxiv.org/abs/2402.17676 |