On the free boundary for thin obstacle problems with Sobolev variable coefficients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914882526380032 |
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| author | Andreucci, Giovanna Focardi, Matteo Spadaro, Emanuele |
| author_facet | Andreucci, Giovanna Focardi, Matteo Spadaro, Emanuele |
| contents | We establish a quasi-monotonicity formula {for an intrinsic frequency function related to solutions to} thin obstacle problems with zero obstacle driven by quadratic energies with Sobolev $W^{1,p}$ coefficients, with $p$ bigger than the space dimension. From this we deduce several regularity and structural properties of the corresponding free boundaries at those distinguished points with finite order of contact with the obstacle. In particular, we prove the rectifiability {and the local finiteness of the Minkowski content} of the whole free boundary in the case of Lipschitz coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16211 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the free boundary for thin obstacle problems with Sobolev variable coefficients Andreucci, Giovanna Focardi, Matteo Spadaro, Emanuele Analysis of PDEs We establish a quasi-monotonicity formula {for an intrinsic frequency function related to solutions to} thin obstacle problems with zero obstacle driven by quadratic energies with Sobolev $W^{1,p}$ coefficients, with $p$ bigger than the space dimension. From this we deduce several regularity and structural properties of the corresponding free boundaries at those distinguished points with finite order of contact with the obstacle. In particular, we prove the rectifiability {and the local finiteness of the Minkowski content} of the whole free boundary in the case of Lipschitz coefficients. |
| title | On the free boundary for thin obstacle problems with Sobolev variable coefficients |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.16211 |