$C^\infty$ partial regularity of the singular set in the obstacle problem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866916527164358656 |
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| author | Franceschini, Federico Zatoń, Wiktoria |
| author_facet | Franceschini, Federico Zatoń, Wiktoria |
| contents | We show that the singular set $Σ$ in the classical obstacle problem can be locally covered by a $C^\infty$ hypersurface, up to an "exceptional" set $E$, which has Hausdorff dimension at most $n-2$ (countable, in the $n=2$ case). Outside this exceptional set, the solution admits a polynomial expansion of arbitrarily large order. We also prove that $Σ\setminus E$ is extremely unstable with respect to monotone perturbations of the boundary datum. We apply this result to the planar Hele-Shaw flow, showing that the free boundary can have singular points for at most countable many times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_00923 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $C^\infty$ partial regularity of the singular set in the obstacle problem Franceschini, Federico Zatoń, Wiktoria Analysis of PDEs We show that the singular set $Σ$ in the classical obstacle problem can be locally covered by a $C^\infty$ hypersurface, up to an "exceptional" set $E$, which has Hausdorff dimension at most $n-2$ (countable, in the $n=2$ case). Outside this exceptional set, the solution admits a polynomial expansion of arbitrarily large order. We also prove that $Σ\setminus E$ is extremely unstable with respect to monotone perturbations of the boundary datum. We apply this result to the planar Hele-Shaw flow, showing that the free boundary can have singular points for at most countable many times. |
| title | $C^\infty$ partial regularity of the singular set in the obstacle problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2102.00923 |