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: aligned F (, ) = ₖ^₁, ₏ () { \0\ } _ F () ᵖ dx + | |㵵 ₅ (_) d{ {H}^N-1 } _ | |ᵖ dx, 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 ]1, +, p ∈ 1, + ∞ [, Ω is a bounded, convex domain in {R}^N, R N, ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on {R}^N. R N. We show an upper bound for ₅ (, ) λ F (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 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 aligned ₚ (, ) ^p-1 = ₀ₑₑ₀ₘ₂ W^{1, p () \0\ array} (_ | | \, dx) ᵖ _ F () ᵖ dx+ | |ᵖ F () d{ {H}^N-1 } 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 >0. β > 0. The obtained results are new also in the case of the classical Euclidean Laplacian.