On the first eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian

Abstract

Let Λp p be the best Sobolev embedding constant of W1,p(Ω) ↪ Lp(∂Ω), where Ω is a smooth bounded domain in ℝN. We prove that as p → ∞ the sequence Λp converges to a constant independent of the shape and the volume of Ω, namely 1. Moreover, for any sequence of eigenfunctions up (associated with Λp), normalized by ∥up∥L∞(∂Ω) = 1, there is a subsequence converging to a limit function u∞ which satisfies, in the viscosity sense, an ∞-Laplacian equation with a boundary condition.