Pólya-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: (Formula presented.) 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 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 (Formula presented.) when β>0. The obtained results are new also in the case of the classical Euclidean Laplacian.