On asymptotic expansions for the fractional infinity Laplacian

Abstract

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional ∞-Laplacian Δ ∞ s for s ∈ ( 1 2 , 1 ). This operator has been introduced and first studied in (Comm. Pure Appl. Math. 65 (2012) 337–380). Our expansions are parametrised by the radius of the removed singularity ε, and allow for the identification of Δ ∞ s ϕ ( x ) as the ε 2 s -order coefficient of the deviation of the ε-average from the value ϕ ( x ), in the limit ε → 0 +. The averages are well posed for functions ϕ that are only Borel regular and bounded.