Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909337563168768 |
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| author | Della Pietra, Francesco Piscitelli, Gianpaolo |
| author_facet | Della Pietra, Francesco Piscitelli, Gianpaolo |
| contents | The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation*} λ_1(β,Ω)=\min_{ψ\in W^{1,p}(Ω)\setminus\{0\} } \frac{\displaystyle\int_ΩF(\nabla ψ)^p dx +β\displaystyle\int_{\partialΩ}|ψ|^pF(ν_Ω) d\mathcal H^{N-1} }{\displaystyle\int_Ω|ψ|^p dx}, \end{equation*} where $p\in]1,+\infty[$, $Ω$ is a bounded, mean convex domain in $\mathbb R^{N}$, $ν_Ω$ is its Euclidean outward normal, $β$ is a real number, and $F$ is a sufficiently smooth norm on $\mathbb R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $β$ and on geometrical quantities associated to $Ω$. More precisely, we prove a lower bound of $λ_{1}$ in the case $β>0$, and a upper bound in the case $β<0$. As a consequence, we prove, for $β>0$, a lower bound for $λ_{1}(β,Ω)$ in terms of the anisotropic inradius of $Ω$ and, for $β<0$, an upper bound of $λ_{1}(β,Ω)$ in terms of $β$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_01814 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators Della Pietra, Francesco Piscitelli, Gianpaolo Analysis of PDEs 35J25 35P15 47J10 47J30 The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation*} λ_1(β,Ω)=\min_{ψ\in W^{1,p}(Ω)\setminus\{0\} } \frac{\displaystyle\int_ΩF(\nabla ψ)^p dx +β\displaystyle\int_{\partialΩ}|ψ|^pF(ν_Ω) d\mathcal H^{N-1} }{\displaystyle\int_Ω|ψ|^p dx}, \end{equation*} where $p\in]1,+\infty[$, $Ω$ is a bounded, mean convex domain in $\mathbb R^{N}$, $ν_Ω$ is its Euclidean outward normal, $β$ is a real number, and $F$ is a sufficiently smooth norm on $\mathbb R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $β$ and on geometrical quantities associated to $Ω$. More precisely, we prove a lower bound of $λ_{1}$ in the case $β>0$, and a upper bound in the case $β<0$. As a consequence, we prove, for $β>0$, a lower bound for $λ_{1}(β,Ω)$ in terms of the anisotropic inradius of $Ω$ and, for $β<0$, an upper bound of $λ_{1}(β,Ω)$ in terms of $β$. |
| title | Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators |
| topic | Analysis of PDEs 35J25 35P15 47J10 47J30 |
| url | https://arxiv.org/abs/2204.01814 |