On comparison of fractional Laplacians

TL;DR

This paper compares two natural types of fractional Laplacians, the restricted Dirichlet and spectral Neumann, revealing how their difference's quadratic form behaves depending on the fractional order and domain convexity.

Contribution

It provides a detailed comparison of the restricted Dirichlet and spectral Neumann fractional Laplacians, including sign behavior of their difference and positivity preservation in convex domains.

Findings

Quadratic form of the difference depends on the parity of the integer part of s.

Difference is positivity preserving for s in (0,1) on convex domains.

Sign of the quadratic form alternates with the parity of the integer part of s.

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.