Pólya-type estimates for the first Robin eigenvalue of elliptic operators

Abstract

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 $p\in ]1,+\infty [$, $\hat{I}\text{c◯}$ is a bounded, convex domain in ${\mathbb{R}}^N$, $\hat{I}{½}_{\hat{I}}\text{c◯}$ is its Euclidean outward normal, ${\hat{I}}^2$ is a real number, and $F$ is a sufficiently smooth norm on ${\mathbb{R}}^N$. We show an upper bound for $\hat{I}{»}_F({\hat{I}}^2,\hat{I}\text{c◯})$ in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on ${\hat{I}}^2$ and on the volume and the anisotropic perimeter of $\hat{I}\text{c◯}$, 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 ${\hat{I}}^2>0$. The obtained results are new also in the case of the classical Euclidean Laplacian.