On the approximation of finite perimeter sets
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917353337389056 |
|---|---|
| author | Carbotti, Alessandro Cito, Simone La Manna, Domenico Angelo Pratelli, Aldo Stefani, Giorgio |
| author_facet | Carbotti, Alessandro Cito, Simone La Manna, Domenico Angelo Pratelli, Aldo Stefani, Giorgio |
| contents | We prove that if $Ω\subseteq\mathbb{R}^N$ is a set with finite perimeter with $\mathscr{H}^{N-1}(\partial Ω\setminus\partial^* Ω)=0$, then any set of finite perimeter $E\subseteq\mathbb{R}^N$ can be approximated by a polyhedral or smooth bounded set $F$ in such a way that both the total perimeter of $E$ and the perimeter of $E$ inside $Ω$ are approximated by those of $F$, and the boundary of $F$ has negligible intersection with the boundary of $Ω$. In addition, we address the approximation for perimeter and volume with densities, and we present counterexamples illustrating the sharpness of our assumptions. Our constructions rely on a technical result that replaces $E$ with a set $F$ which agrees with $E$ and has the same boundary inside $Ω$, while sharing no common boundary with $Ω$, and does so without substantially altering the perimeter or the volume of the original set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18984 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the approximation of finite perimeter sets Carbotti, Alessandro Cito, Simone La Manna, Domenico Angelo Pratelli, Aldo Stefani, Giorgio Functional Analysis We prove that if $Ω\subseteq\mathbb{R}^N$ is a set with finite perimeter with $\mathscr{H}^{N-1}(\partial Ω\setminus\partial^* Ω)=0$, then any set of finite perimeter $E\subseteq\mathbb{R}^N$ can be approximated by a polyhedral or smooth bounded set $F$ in such a way that both the total perimeter of $E$ and the perimeter of $E$ inside $Ω$ are approximated by those of $F$, and the boundary of $F$ has negligible intersection with the boundary of $Ω$. In addition, we address the approximation for perimeter and volume with densities, and we present counterexamples illustrating the sharpness of our assumptions. Our constructions rely on a technical result that replaces $E$ with a set $F$ which agrees with $E$ and has the same boundary inside $Ω$, while sharing no common boundary with $Ω$, and does so without substantially altering the perimeter or the volume of the original set. |
| title | On the approximation of finite perimeter sets |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2603.18984 |