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

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: $${\lambda }_F(\beta ,\mathrm{\Omega })={\lambda }_F(p,\beta ,\mathrm{\Omega })=\underset{\psi \in W^{1,p}(\mathrm{\Omega })\setminus \{0\}}{\min }\frac{{\int }_{\mathrm{\Omega }}F(\mathrm{\nabla }\psi {)}^{p}dx+\beta {\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }\psi {\mathrm{\mid }}^{p}F({\nu }_{\mathrm{\Omega }})d{\mathcal{H}}^{N-1}}{{\int }_{\mathrm{\Omega }}\mathrm{\mid }\psi {\mathrm{\mid }}^{p}dx}$$ where $p\in ]1,+\mathrm{\infty }[$, $\mathrm{\Omega }$ is a bounded, convex domain in ${\mathbb{R}}^N$, ${\nu }_{\mathrm{\Omega }}$ is its Euclidean outward normal, $\beta$ is a real number, and $F$ is a sufficiently smooth norm on ${\mathbb{R}}^N$. We show an upper bound for ${\lambda }_F(\beta ,\mathrm{\Omega })$ in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $\beta$ and on the volume and the anisotropic perimeter of $\mathrm{\Omega }$, 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 $${\tau }_p(\beta ,\mathrm{\Omega }{)}^{p-1}=\underset{\begin{array}{c}\psi \in W^{1,p}(\mathrm{\Omega })\setminus \{0\}\end{array}}{\max }{\displaystyle \frac{{\left({\int }_{\mathrm{\Omega }}\mathrm{\mid }\psi \mathrm{\mid }dx\right)}^p}{{\int }_{\mathrm{\Omega }}F(\mathrm{\nabla }\psi {)}^{p}dx+\beta {\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }\psi {\mathrm{\mid }}^{p}F({\nu }_{\mathrm{\Omega }})d{\mathcal{H}}^{N-1}}},$$ when $\beta >0$. The obtained results are new also in the case of the classical Euclidean Laplacian.