Rayleigh–Faber–Krahn, Lyapunov and Hartmann–Wintner Inequalities for 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.