On asymptotic expansions for the fractional infinity Laplacian

TL;DR

This paper develops two asymptotic expansions for the fractional infinity Laplacian using integral averages, enabling the identification of the operator as a specific coefficient in the limit of shrinking singularity radius.

Contribution

It introduces novel asymptotic expansions for the fractional infinity Laplacian based on integral averages, applicable to functions with minimal regularity.

Findings

Identified the fractional infinity Laplacian as an epsilon^{2s}-order coefficient.

Provided well-posed integral averages for Borel regular and bounded functions.

Established the limit behavior of the averages as epsilon approaches zero.

Abstract

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional $\infty$-Laplacian $\Delta_\infty^s$ for $s\in (\frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland, C., Caffarelli, L. and Figalli, A., \textsl{Nonlocal Tug-of-War and the inifnity fractional Laplacian}, Comm. Pure Appl. Math., \textbf{65}, pp. 337--380, (2012)]. Our expansions are parametrised by the radius of the removed singularity $\epsilon$, and allow for the identification of $\Delta_\infty^s\phi(x)$ as the $\epsilon^{2s}$-order coefficient of the deviation of the $\epsilon$-average from the value $\phi(x)$, in the limit $\epsilon\to 0+$. The averages are well posed for functions $\phi$ that are only Borel regular and bounded.