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 $\infty$-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 ${\Delta }_{\infty }^s\varphi (x)$ as the ${\epsilon }^{2s}$-order coefficient of the deviation of the $\epsilon$-average from the value $\varphi (x)$, in the limit $\epsilon \to 0+$. The averages are well posed for functions $\varphi$ that are only Borel regular and bounded.

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References in corpus (1)

• An asymptotic expansion for the fractional $p$-Laplacian and for gradient dependent nonlocal operators