The ∞-Eigenvalue Problem

Asymptotics for the best Sobolev constants and their extremal functions

“…(Ω). This property is also shared with the distance function ρ and the first eigenfunctions of the ∞-Laplacian (see [12]). In the sequel (see Theorem 3.14) we will prove that u ∞ = ρ ρ ∞ for some special domains.…”

“…We use the bounds (12) and (9) and the explicit expression of S p in (5) to conclude that the function p → Λ p (Ω) is continuous at p = N.…”

The Dependence of the First Eigenvalue of the Infinity Laplacian With Respect to the Domain

“…The limit as p → ∞ of the eigenvalue problem was studied in [13], [12] and an anisotropic version in [4]. In those papers the authors prove that…”

“…These extremals can be constructed taking the limit as p → ∞ in the eigenfunctions of the p−laplacian eigenvalue problem (see [12]) and are viscosity solutions of the following eigenvalue problem (called the infinity eigenvalue problem in the literature):…”

“…By looking at the variational quotient that defines λ 1 (Ω), in [12] it is also proved that the first eigenvalue has a simple geometrical characterization, it holds that…”