On the shape of the first fractional eigenfunction

TL;DR

This paper investigates the properties of the first eigenfunction of the fractional Laplacian for s in (1/2,1), demonstrating its superharmonicity in the unit ball up to dimension 11, using analytical and computer-assisted methods.

Contribution

It establishes superharmonicity of the first fractional eigenfunction in certain dimensions and introduces a computer-assisted approach to estimate complex constants.

Findings

First eigenfunction is superharmonic in the unit ball for dimensions up to 11.

A computer-assisted method is used to estimate key constants.

Provides new insights into fractional Laplacian eigenfunctions.

Abstract

We show that the first eigenfunction of the fractional Laplacian ${(-\Delta)}^s$, $s\in(1/2,1)$, is superharmonic in the unitary ball up to dimension $11$. To this aim, we also rely on a computer-assisted step to estimate a rather complicated constant depending on the dimension and the power $s$.