### 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 e i, λ i are the eigenfunctions and the eigenvalues of the Laplace operator -Δ in Ω with homogeneous Dirichlet boundary data, while a i represents the projection of u on the direction e i. The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

Original language | English |
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Pages (from-to) | 831-855 |

Number of pages | 25 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 144 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jan 2014 |

Externally published | Yes |