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:

$$\begin{aligned} \lambda _F(\beta ,\Omega )= \min _{\psi \in W^{1,p}(\Omega ){\setminus }\{0\} } \frac{\displaystyle \int _\Omega F(\nabla \psi )^p dx +\beta \int _{\partial \Omega }|\psi |^p F(\nu _{\Omega }) d{\mathcal {H}}^{N-1} }{\displaystyle \int _\Omega |\psi |^p dx}, \end{aligned}$$

where \(p\in ]1,+\infty [,\) \(\Omega \) is a bounded, convex domain in \({\mathbb {R}}^{N},\) \(\nu _{\Omega }\) 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 \(\lambda _{F}(\beta ,\Omega )\) 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 \(\Omega ,\) in the spirit of the classical estimates of Pólya (J Indian Math Soc (NS) 24:413–419, 1961) for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity

$$\begin{aligned} \tau _p(\beta ,\Omega )^{p-1} = \max _{\begin{array}{c} \psi \in W^{1,p}(\Omega ){\setminus }\{0\} \end{array}} \dfrac{\left( \displaystyle \int _\Omega |\psi | \, dx\right) ^p}{\displaystyle \int _\Omega F(\nabla \psi )^p dx+\beta \int _{\partial \Omega }|\psi |^p F(\nu _{\Omega }) d{\mathcal {H}}^{N-1} } \end{aligned}$$

when \(\beta >0.\) The obtained results are new also in the case of the classical Euclidean Laplacian.