On the spectrum of two different fractional operators

Abstract

Abstract. In this paper we deal with two nonlocal operators, that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. Precisely, for a fixed s ∈ (0, 1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that iswhere ei , λi are the eigenfunctions and the eigenvalues of the Laplace operator −∆ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei . Aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

The Brezis–Nirenberg problem for nonlocal systems

“…in [2]. See [14] for a nice comparison between these two different notions of fractional laplacian in bounded domains. By choosing p = q = 2 * s /2, system (1.4) reduces to…”

Density properties for fractional Sobolev spaces

“…Recently, in the literature a deep interest was shown for nonlocal operators, thanks to their intriguing analytical structure and in view of several applications in a wide range of contexts, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science and water waves: see for instance [3,4,5,6,8,9,10,11,14,15,16,20,21,22,23,24,27,29,30,31,32,33] and references therein. One of the typical models considered is the equation may be defined as (1.2) −(−∆) s u(x) =ˆR n u(x + y) + u(x − y) − 2u(x) |y| n+2s dy for x ∈ R n (see [12,28] and references therein for further details on the fractional Laplacian) and the right-hand side f is a function satisfying suitable regularity and growth conditions. Problem (1.1) has a variational structure and the natural space where finding solutions for it is the fractional Sobolev space H s (R n ) (see [1,12]).…”

Models of space-fractional diffusion: A critical review

“…This setting shows clearly that for non-local differential operators, initial or boundary data should be given also outside of the computational domain Ω. For a spectral comparison of the spatial operator in (8) with the one in (7), we refer to [19].…”