On comparison of fractional Laplacians

Abstract

For $s>-1$, $s\notin\mathbb N_0$, we compare two natural types of fractional Laplacians $(-\Delta)^s$, namely, the restricted Dirichlet and the spectral Neumann ones. We show that for the quadratic form of their difference taken on the space $\tilde{H}^s(\Omega)$ is positive or negative depending on whether the integer part of $s$ is even or odd. For $s\in(0,1)$ and convex domains we prove also that the difference of these operators is positivity preserving on $\tilde{H}^s(\Omega)$. This paper complements [10] and [11] where similar statements were proved for the spectral Dirichlet and the restricted Dirichlet fractional Laplacians.