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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.11876 |
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| _version_ | 1866908322203959296 |
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| author | Colombo, Giacomo |
| author_facet | Colombo, Giacomo |
| contents | We show that the free boundary of a solution of the Stefan problem in $\mathbb R^{4+1}$ is a $3$-dimensional manifold of class $C^\infty$ in $\mathbb R^4$ for almost every time. This is achieved by showing that for all dimensions $n$ the singular set $Σ\subset \mathbb R^{n+1}$ can be decomposed in two parts $Σ=Σ^\infty\cup Σ^*$, where $Σ^\infty$ is covered by one $(n-1)$-dimensional manifold of class $C^\infty$ in $\mathbb R^{n+1}$ and its projection onto the time axis has Hausdorff dimension 0, while $Σ^*$ is parabolically countably $(n-2)$-rectifiable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11876 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generic regularity in time for solutions of the Stefan problem in 4+1 dimensions Colombo, Giacomo Analysis of PDEs 35R35, 35B65 We show that the free boundary of a solution of the Stefan problem in $\mathbb R^{4+1}$ is a $3$-dimensional manifold of class $C^\infty$ in $\mathbb R^4$ for almost every time. This is achieved by showing that for all dimensions $n$ the singular set $Σ\subset \mathbb R^{n+1}$ can be decomposed in two parts $Σ=Σ^\infty\cup Σ^*$, where $Σ^\infty$ is covered by one $(n-1)$-dimensional manifold of class $C^\infty$ in $\mathbb R^{n+1}$ and its projection onto the time axis has Hausdorff dimension 0, while $Σ^*$ is parabolically countably $(n-2)$-rectifiable. |
| title | Generic regularity in time for solutions of the Stefan problem in 4+1 dimensions |
| topic | Analysis of PDEs 35R35, 35B65 |
| url | https://arxiv.org/abs/2504.11876 |