Minimally singular functions and the rigidity problem for Steiner's perimeter inequality
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909405744726016 |
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| author | Perugini, Matteo |
| author_facet | Perugini, Matteo |
| contents | Let $n\geq 1$, and let $Ω\subset \mathbb{R}^n$ be an open and connected set with finite Lebesgue measure. Among functions of bounded variation in $Ω$ we introduce the class of \emph{minimally singular} functions. Inspired by the original theory of Vol'pert of one-dimensional restrictions of $BV$ functions, we provide a geometric characterization for this class of functions via the introduction of a pseudometric that we call \emph{singular vertical distance}. As an application, we present a characterization result for \emph{rigidity} of equality cases for Steiner's perimeter inequality. By \emph{rigidity} we mean that the only extremals for Steiner's perimeter inequality are vertical translations of the Steiner symmetric set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimally singular functions and the rigidity problem for Steiner's perimeter inequality Perugini, Matteo Analysis of PDEs 49Q20, 49Q10 Let $n\geq 1$, and let $Ω\subset \mathbb{R}^n$ be an open and connected set with finite Lebesgue measure. Among functions of bounded variation in $Ω$ we introduce the class of \emph{minimally singular} functions. Inspired by the original theory of Vol'pert of one-dimensional restrictions of $BV$ functions, we provide a geometric characterization for this class of functions via the introduction of a pseudometric that we call \emph{singular vertical distance}. As an application, we present a characterization result for \emph{rigidity} of equality cases for Steiner's perimeter inequality. By \emph{rigidity} we mean that the only extremals for Steiner's perimeter inequality are vertical translations of the Steiner symmetric set. |
| title | Minimally singular functions and the rigidity problem for Steiner's perimeter inequality |
| topic | Analysis of PDEs 49Q20, 49Q10 |
| url | https://arxiv.org/abs/2411.17633 |