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

TL;DR

The paper extends classical eigenvalue inequalities to a fractional elliptic operator in cylindrical domains $D=(a,b)\times\Omega$, combining left-Riemann–Liouville and right-Caputo derivatives in the axial direction with a fractional Laplacian in the cross-section. By separating variables and leveraging symmetric rearrangements (circular and polygonal cross-sections), it proves Rayleigh-Faber-Krahn-type inequalities: the first eigenvalue $\nu_1(D)$ is minimized by circular cylinders (and by equilateral triangles or squares in polygonal cases) among domains of fixed measure. It also derives Lyapunov and Hartman–Wintner inequalities for the fractional problem by reducing to a one-dimensional scalar problem via projection onto the first cross-sectional eigenfunction and exploiting Green’s function kernels. These results generalize classical spectral inequalities to fractional order operators, offering sharp bounds and variational characterizations relevant to fractional PDEs with Dirichlet boundary conditions.

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.

Theorems & Definitions (23)

• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• Theorem 3.3
• proof
• Corollary 3.4