Abstract:
For non-integer $s>-1$, we compare two natural types of fractional Laplacians $(-\Delta)^s$, namely, the restricted Dirichlet and the spectral Neumann ones. We show that the quadratic form of their difference taken on the space
$\widetilde H^s(\Omega)$ 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 $\widetilde H^s(\Omega)$. This paper complements earlier works of R. Musina and author where similar statements were proved for the spectral Dirichlet and the restricted Dirichlet fractional Laplacians.
