Pólya-type estimates for the first Robin eigenvalue of elliptic operators
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911776305577984 |
|---|---|
| author | Della Pietra, F. |
| author_facet | Della Pietra, F. |
| contents | The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \[
λ_F(β,Ω)=λ_{F}(p,β,Ω)= \min_{ψ\in W^{1,p}(Ω)\setminus\{0\} } \frac{\int_ΩF(\nabla ψ)^p dx +β\int_{\partialΩ}|ψ|^p F(ν_Ω) d\mathcal H^{N-1} }{\int_Ω|ψ|^p dx} \] where $p\in]1,+\infty[$, $Ω$ is a bounded, 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 $\mathbb R^{N}$. We show an upper bound for $λ_{F}(β,Ω)$ in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $β$ and on the volume and the anisotropic perimeter of $Ω$, in the spirit of the classical estimates of Pólya \cite{po61} for the Euclidean Dirichlet Laplacian.
We will also provide a lower bound for the torsional rigidity \[ τ_p(β,Ω)^{p-1} = \max_{\substack{ψ\in
W^{1,p}(Ω)\setminus\{0\}}} \dfrac{\left(\int_Ω|ψ| \,
dx\right)^p}{\int_ΩF(\nablaψ)^p dx+β\int_{\partialΩ}|ψ|^p F(ν_Ω) d\mathcal H^{N-1} }, \] when $β>0$. The obtained results are new also in the case of the classical Euclidean Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08474 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pólya-type estimates for the first Robin eigenvalue of elliptic operators Della Pietra, F. Analysis of PDEs The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \[ λ_F(β,Ω)=λ_{F}(p,β,Ω)= \min_{ψ\in W^{1,p}(Ω)\setminus\{0\} } \frac{\int_ΩF(\nabla ψ)^p dx +β\int_{\partialΩ}|ψ|^p F(ν_Ω) d\mathcal H^{N-1} }{\int_Ω|ψ|^p dx} \] where $p\in]1,+\infty[$, $Ω$ is a bounded, 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 $\mathbb R^{N}$. We show an upper bound for $λ_{F}(β,Ω)$ in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $β$ and on the volume and the anisotropic perimeter of $Ω$, in the spirit of the classical estimates of Pólya \cite{po61} for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity \[ τ_p(β,Ω)^{p-1} = \max_{\substack{ψ\in W^{1,p}(Ω)\setminus\{0\}}} \dfrac{\left(\int_Ω|ψ| \, dx\right)^p}{\int_ΩF(\nablaψ)^p dx+β\int_{\partialΩ}|ψ|^p F(ν_Ω) d\mathcal H^{N-1} }, \] when $β>0$. The obtained results are new also in the case of the classical Euclidean Laplacian. |
| title | Pólya-type estimates for the first Robin eigenvalue of elliptic operators |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.08474 |