Sharp estimates for the first Robin eigenvalue of nonlinear 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:λ1(β,Ω)=minψ∈W1,p(Ω)∖{0}⁡∫ΩF(∇ψ)pdx+β∫∂Ω|ψ|pF(νΩ)dHN−1∫Ω|ψ|pdx, where p∈]1,+∞[, Ω is a bounded, anisotropic mean convex domain in RN, νΩ is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on RN. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ1 in the case β>0, and a upper bound in the case β<0. As a consequence, we prove, for β>0, a lower bound for λ1(β,Ω) in terms of the anisotropic inradius of Ω and, for β<0, an upper bound of λ1(β,Ω) in terms of β.