The eigenvalue problem for the regional fractional Laplacian in the small order limit

TL;DR

This paper investigates the asymptotic behavior of eigenvalues and eigenfunctions of the regional fractional Laplacian as the order approaches zero, introducing the regional logarithmic Laplacian as a key concept.

Contribution

It provides a novel analysis of the eigenvalue problem for the regional fractional Laplacian in the small order limit, connecting it to the regional logarithmic Laplacian.

Findings

Eigenvalues and eigenfunctions exhibit specific asymptotic behavior as s approaches 0

Introduction of the regional logarithmic Laplacian as a derivative at s=0

Insights into the spectral properties of fractional Laplacians in the small order regime

Abstract

In this note, we study the asymptotic behavior of eigenvalues and eigenfunctions of the regional fractional Laplacian $(-\Delta)^s$ as $ s \to 0^+$. Our analysis leads to a study of the regional logarithmic Laplacian, which arises as a formal derivative of regional fractional Laplacians at $s = 0$.