On asymptotic expansions for the fractional infinity Laplacian

Article type: Research Article

Authors: del Teso, Félix^{a; *} | Endal, Jørgen^b | Lewicka, Marta^c

Affiliations: [a] Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain. E-mail: fdelteso@ucm.es | [b] Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain. E-mail: jorgen.endal@uam.es | [c] Department of Mathematics, University of Pittsburgh, 139 University Place, Pittsburgh, PA 15260, USA. E-mail: lewicka@pitt.edu

Correspondence: [*] Corresponding author. E-mail: fdelteso@ucm.es.

Abstract: We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional ∞-Laplacian Δ∞s for s∈(12,1). 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∞sϕ(x) as the ε2s-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.

Keywords: Mean value formulas, nonlocal gradient dependent operators, fractional infinity Laplacian

DOI: 10.3233/ASY-211686

Journal: Asymptotic Analysis, vol. 127, no. 3, pp. 201-216, 2022