Fractional calculus for power functions

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.

18 pages, added Proposition 2 and references, some minor corrections

References in corpus (2)

• Spectral properties of the Cauchy process
• Barenblatt profiles for a nonlocal porous media equation

Cited by in corpus (3)

• Some observations on the Green function for the ball in the fractional Laplace framework
• Fractional Laplacians in bounded domains: Killed, reflected, censored and taboo Lévy flights
• Levy flights in steep potential wells: Langevin modeling versus direct response to energy landscapes