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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2017
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| Accès en ligne: | https://arxiv.org/abs/1704.00508 |
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| _version_ | 1866910635445452800 |
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| author | Della Pietra, Francesco Gavitone, Nunzia Piscitelli, Gianpaolo |
| author_facet | Della Pietra, Francesco Gavitone, Nunzia Piscitelli, Gianpaolo |
| contents | Let $Ω$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $λ_{2}(p,Ω)$ of the anisotropic $p$-Laplacian \[ -\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $λ_{2}(p,Ω)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1704_00508 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators Della Pietra, Francesco Gavitone, Nunzia Piscitelli, Gianpaolo Analysis of PDEs 35P15 - 35P30 - 35J60 Let $Ω$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $λ_{2}(p,Ω)$ of the anisotropic $p$-Laplacian \[ -\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $λ_{2}(p,Ω)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$. |
| title | On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators |
| topic | Analysis of PDEs 35P15 - 35P30 - 35J60 |
| url | https://arxiv.org/abs/1704.00508 |