Asymptotic mean value properties for fractional anisotropic operators

Abstract

We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain-Brezis-Mironescu type limit formula for a related class of anisotropic nonlocal norms.2010 Mathematics Subject Classification. 46E35, 28D20, 82B10, 49A50. Key words and phrases. Mean value formulas, anisotropic fractional operators. The second author is member of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games

“…To our knowledge, the operators H s,± ∞ have previously not been discussed in literature, but we note that when applied to functions which are independent of time, then the operators formally coincide with −∆ s ∞ , where ∆ s ∞ is the infinity fractional Laplacian introduced in [8]. We note that in the stationary case, versions of Theorem 2.2 are proved in [9].…”

Essentials of Nonlocal Operators

“…The main result relative to the fractional p-Laplacian, that we prove in Section 2 (Theorem 2.7), is the following. Notice that for p = 2 the result in [6] is recovered. In the case we consider here, however, the dependence of D s,p r of the function u does not allow a simplification of the formula obtained.…”

“…The equivalence between s-harmonic functions and the fractional mean value property is proved in [1] (see also [11], [5]), with the fractional mean kernel given by (1.3) M s r u(x) = c(n, s)r 2s R n \Br u(x − y) (|y| 2 − r 2 ) s |y| n dy, where c(n, s) = Γ(n/2) sin πs/π n/2+1 . Furthermore, in [6] the authors obtain an asymptotic expansion for harmonic functions with respect to a fractional anisotropic operator (that includes the case of the fractional Laplacin). Precisely, a continuous function u is harmonic in the viscosity sense if and only if (1.3) holds in a viscosity sense up to a rest of order two, namely (1.4) u(x) = c(n, s)r 2s R n \Br u(x − y) (|y| 2 − r 2 ) s |y| n dy + o(r 2 ), The goal of this paper is to continue the analysis of the nonlocal case and to provide a nonlocal counterpart (in some sense) of the result by Manfredi, Parviainen and Rossi [14] for the (s, p)-Laplacian (−∆) s p .…”

“…R n \Br u(x − y) (|y| 2 − r 2 ) s |y| n dy, where c(n, s) = Γ(n/2) sin πs/π n/2+1 . Furthermore, in [6] the authors obtain an asymptotic expansion for harmonic functions with respect to a fractional anisotropic operator (that includes the case of the fractional Laplacin). Precisely, a continuous function u is harmonic in the viscosity sense if and only if (1.3) holds in a viscosity sense up to a rest of order two, namely (1.4) u(x) = c(n, s)r 2s…”

Non-local tug-of-war with noise for the geometric fractional $p$-Laplacian

“…Asymptotic expansions for gradient-dependent operators have been recently discussed in [3,4]. However, the averages in there depended on ∇u(x), which is a drawback in the context of our further applications, based on solutions to the truncated expansions (DPP) ε .…”