Rayleigh-Faber-Krahn, Lyapunov and Hartmann-Wintner inequalities for fractional elliptic problems

TL;DR

This paper establishes that the first eigenvalue of a fractional elliptic operator in cylindrical domains is minimized in a circular cylinder of the same volume, and extends Lyapunov and Hartmann-Wintner inequalities to fractional elliptic problems.

Contribution

It proves a fractional Rayleigh-Faber-Krahn inequality and extends Lyapunov and Hartmann-Wintner inequalities to fractional elliptic boundary value problems.

Findings

First eigenvalue minimized in circular cylinders of same volume

Extension of Lyapunov inequalities to fractional elliptic problems

Extension of Hartmann-Wintner inequalities to fractional elliptic problems

Abstract

In this paper in the cylindrical domain we consider a fractional elliptic operator with Dirichlet conditions. We prove, that the first eigenvalue of the fractional elliptic operator is minimised in a circular cylinder among all cylindrical domains of the same Lebesgue measure. This inequality is called the Rayleigh-Faber-Krahn inequality. Also, we give Lyapunov and Hartmann-Wintner inequalities for the fractional elliptic boundary value problem.