In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operator Δpu=∑i∂∂xi|∇u|p−2-0.16em∂u∂xi, that is the p‐Laplacian, and trueΔ̃pu=∑i∂∂xi|∂u∂xi|p−2∂u∂xi, namely the pseudo‐p‐Laplacian. Moreover we prove a stability result by means of a suitable isoperimetric deficit. Finally, we give a sharp lower bound for the anisotropic p‐torsional rigidity.

This paper is devoted to the study of a quantitative Weinstock inequality in higher dimension for the first non trivial Steklov eigenvalue of the Laplace operator for convex sets.The key role is played by a quantitative isoperimetric inequality which involves the boundary momentum, the volume and the perimeter of a convex open set of R n , n ≥ 2.

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In this paper we consider the eigenvalue problem for a fully nonlinear equation involving Hessian operators. In particular we study some properties of the first eigenvalue and of corresponding eigenfunctions. Using suitable symmetrization arguments, we prove a Faber-Krahn inequality for the first eigenvalue and a PayneRayner type inequality for eigenfunctions, which are well known for the p-laplacian operator and the Monge-Ampere operator.

We study some Hardy-type inequalities involving a general norm in ℝn and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.