12-123 Raffaella Servadei, Enrico Valdinoci

On the spectrum of two different fractional operators (539K, pdf) Oct 19, 12

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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 $s\in (0,1)$ we consider the \emph{integral} definition of the fractional Laplacian given by $$(-\Delta)^s u(x):=rac{c(n,s)}{2}\int_{\RR^{n}}rac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\,dy\,,\,\,\,\, x\in \RR^n\,,$$ where $c(n,s)$ is a positive normalizing constant, and another fractional operator obtained via a \emph{spectral} definition, that is $$A_s u=\sum_{i\in \mathbb N}a_i\,\lambda_i^s\,e_i\,,$$ where $e_i\,, \lambda_i$ are the eigenfunctions and the eigenvalues of the Laplace operator $-\Delta$ in $\Omega$ with homogeneous Dirichlet boundary data, while $a_i$ represents the projection of $u$ on the direction $e_i$\,. Aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

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