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On the spectrum of two different fractional operators

Published online by Cambridge University Press:  24 July 2014

Raffaella Servadei  and
Enrico Valdinoci
Raffaella Servadei
Affiliation:
Dipartimento di Matematica e Informatica, Università della Calabria, Ponte Pietro Bucci 31 B, 87036 Arcavacata di Rende (Cosenza), Italy, [email protected]
Enrico Valdinoci
Affiliation:
Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin, Germany, Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133 Milano, Italy and Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy, [email protected]
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Abstract

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given by

where c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,

where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.

The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 4 , August 2014 , pp. 831 - 855
Copyright
Copyright © Royal Society of Edinburgh 2014 

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