P\'olya-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: \[ λ_F(β,Ω)=λF(p,β,Ω)= min_{ψ∈ W1,p(Ω)\{0\} } {∫_Ω F(∇ ψ)^p dx +β∫∂Ω|ψ|^p F(νΩ) d HN-1 }{∫_Ω|ψ|^p dx} \] where p∈]1,+∞[, Ω is a bounded, convex domain in RN, νΩ is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on RN. We show an upper bound for λF(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 P\'olya {po61} for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity \[ τ_p(β,Ω)ᵖ⁻¹ = max_{{ψ∈ W1,p(Ω)\{0\}}} {(∫_Ω |ψ| \, dx)^p}{∫_Ω F(∇ψ)^p dx+β ∫∂Ω|ψ|^p F(νΩ) d HN-1 }, \] when β>0. The obtained results are new also in the case of the classical Euclidean Laplacian.