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

2024/02/13 by F. Della Pietra, Della Pietra, F.
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2402.08474

Abstract

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)=λ_F(p,β,Ω)= min_ψ∈ W^1,p(Ω)∖\0\ \frac∫_ΩF(∇ ψ)^p dx +β∫_∂Ω|ψ|^p F(ν_Ω) d\mathcal H^N-1 ∫_Ω|ψ|^p dx where p∈]1,+∞[, Ω is a bounded, convex domain in \mathbb R^N, ν_Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on \mathbb R^N. We show an upper bound for 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 \citepo61 for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity τ_p(β,Ω)^p-1 = max_\substackψ∈ W^1,p(Ω)∖\0\ \dfrac(∫_Ω|ψ| dx)^p∫_ΩF(∇ψ)^p dx+β∫_∂Ω|ψ|^p F(ν_Ω) d\mathcal H^N-1 , when β>0. The obtained results are new also in the case of the classical Euclidean Laplacian.

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