THE p-FABER-KRAHN INEQUALITY NOTED

When revisiting the Faber-Krahn inequality for the principal pLaplacian eigenvalue of a bounded open set in Rn with smooth boundary, we simply rename it as the p-Faber-Krahn inequality and interestingly find that this inequality may be improved but also characterized through Maz’ya’s capacity method, the Euclidean volume, the Sobolev type inequality and MoserTrudinger’s inequality. 1. The p-Faber-Krahn Inequality Introduced Throughout this article, we always assume that Ω is a bounded open set with smooth boundary ∂Ω in the 2 ≤ n-dimensional Euclidean space R equipped with the scalar product 〈·, ·〉, but also dV and dA stand respectively for the n and n− 1 dimensional Hausdorff measure elements on R. For 1 ≤ p < ∞, the p-Laplacian of a function f on Ω is defined by ∆pf = − div (|∇f | ∇f). As usual, ∇ and div (|∇|∇) mean the gradient and p-harmonic operators respectively (cf. [8]). If W 1,p 0 (Ω) denotes the p-Sobolev space on Ω – the closure of all smooth functions f with compact support in Ω (written as f ∈ C 0 (Ω)) under the norm