A comparison method for the fractional Laplacian and applications

TL;DR

This paper investigates the boundary behavior of solutions to fractional elliptic equations, establishing eigenvalue isolation, a generalized Hopf's lemma, and a boundary Harnack inequality, advancing understanding of fractional Laplacian problems.

Contribution

It introduces new results on eigenvalue isolation, a generalized Hopf's lemma, and a boundary Harnack inequality for fractional elliptic equations, with applications to fractional Laplacian analysis.

Findings

Eigenvalue of fractional Lane-Emden equation is isolated in Wiener regular domains

Established a generalized Hopf's lemma for fractional elliptic equations

Proved a global boundary Harnack inequality for solutions

Abstract

We study the boundary behavior of solutions to fractional elliptic equations. As the first result, the isolation of the first eigenvalue of the fractional Lane-Emden equation is proved in the bounded open sets with Wiener regular boundary. Then, a generalized Hopf's lemma and a global boundary Harnack inequality are proved for the fractional elliptic equations.