On the second anisotropic Cheeger constant and related questions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912300554780672 |
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| author | Piscitelli, Gianpaolo |
| author_facet | Piscitelli, Gianpaolo |
| contents | In this paper we study the behavior of the second eigenfunction of the anisotropic $p$-Laplace operator \[ - Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] as $p \to 1^+$, where $F$ is a suitable smooth norm of $\mathbb R^{n}$. Moreover, for any regular set $Ω$, we define the second anisotropic Cheeger constant as \begin{equation*} h_{2,F}(Ω):=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\},\; E_{1},E_{2}\subset Ω, E_{1}\cap E_{2}=\emptyset\right\}, \end{equation*} where $P_{F}(E)$ is the anisotropic perimeter of $E$, and study the connection with the second eigenvalue of the anisotropic $p$-Laplacian. Finally, we study the twisted anisotropic $q$-Cheeger constant with a volume constraint. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_23092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the second anisotropic Cheeger constant and related questions Piscitelli, Gianpaolo Analysis of PDEs 47J10 - 49Q20 - 52A38 In this paper we study the behavior of the second eigenfunction of the anisotropic $p$-Laplace operator \[ - Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] as $p \to 1^+$, where $F$ is a suitable smooth norm of $\mathbb R^{n}$. Moreover, for any regular set $Ω$, we define the second anisotropic Cheeger constant as \begin{equation*} h_{2,F}(Ω):=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\},\; E_{1},E_{2}\subset Ω, E_{1}\cap E_{2}=\emptyset\right\}, \end{equation*} where $P_{F}(E)$ is the anisotropic perimeter of $E$, and study the connection with the second eigenvalue of the anisotropic $p$-Laplacian. Finally, we study the twisted anisotropic $q$-Cheeger constant with a volume constraint. |
| title | On the second anisotropic Cheeger constant and related questions |
| topic | Analysis of PDEs 47J10 - 49Q20 - 52A38 |
| url | https://arxiv.org/abs/2503.23092 |