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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.06906 |
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| _version_ | 1866917402156990464 |
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| author | Su, Weicong Zhang, Yi Ru-Ya |
| author_facet | Su, Weicong Zhang, Yi Ru-Ya |
| contents | We prove that a trace inequality holds for John domains $Ω$ satisfying $$ \mathcal H^{n-1}(\partial Ω\setminus \partial_*Ω)=0,$$ where $\partial_*Ω$ denotes the measure-theoretic boundary, together with an upper density bound on $\partial Ω$. This class of domains includes $(ε,\,r)$-perimeter minimizers of Wulff perimeter $P_K$ which are close to the associated convex body $K$. Particularly, this result is established without requiring $\partial Ω$ to be Ahlfors regular. As a consequence, we give an alternative proof for a crucial step in the quantitative Wulff inequality, thereby providing a meaningful commentary on the seminal work of Figalli, Maggi, and Pratelli \cite{FMP2010}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06906 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sobolev trace inequalities on John domains and its applications Su, Weicong Zhang, Yi Ru-Ya Optimization and Control 46E35 We prove that a trace inequality holds for John domains $Ω$ satisfying $$ \mathcal H^{n-1}(\partial Ω\setminus \partial_*Ω)=0,$$ where $\partial_*Ω$ denotes the measure-theoretic boundary, together with an upper density bound on $\partial Ω$. This class of domains includes $(ε,\,r)$-perimeter minimizers of Wulff perimeter $P_K$ which are close to the associated convex body $K$. Particularly, this result is established without requiring $\partial Ω$ to be Ahlfors regular. As a consequence, we give an alternative proof for a crucial step in the quantitative Wulff inequality, thereby providing a meaningful commentary on the seminal work of Figalli, Maggi, and Pratelli \cite{FMP2010}. |
| title | Sobolev trace inequalities on John domains and its applications |
| topic | Optimization and Control 46E35 |
| url | https://arxiv.org/abs/2406.06906 |