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

Abstract

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