Fractional calculus for power functions

TL;DR

This paper computes the fractional Laplacian for specific power functions and their derivatives, providing estimates for the first eigenvalues in a ball, advancing understanding of fractional differential operators.

Contribution

It introduces explicit calculations of the fractional Laplacian for particular power functions and applies these to estimate eigenvalues in bounded domains.

Findings

Explicit formulas for fractional Laplacian of power functions

Eigenvalue estimates for fractional Laplacian in a ball

Enhanced understanding of fractional differential operators

Abstract

We calculate the fractional Laplacian for functions of the form $u(x)=(1-|x|^2)_+^p$ and $v(x)=x_d u(x)$. As an application, we estimate the first eigenvalues of the fractional Laplacian in a ball.