The Hadamard formula and the Rayleigh-Faber-Krahn inequality for nonlocal eigenvalue problems

TL;DR

This paper extends classical eigenvalue inequalities and formulas to nonlocal problems, providing a Hadamard formula for eigenvalues under domain perturbations and a Rayleigh-Faber-Krahn inequality using rearrangement techniques.

Contribution

It introduces a Hadamard formula for nonlocal eigenvalues and establishes a Rayleigh-Faber-Krahn inequality for a broad class of nonlocal problems.

Findings

Hadamard formula derived for nonlocal eigenvalues under domain perturbations

Rayleigh-Faber-Krahn inequality proven for nonlocal eigenvalue problems

Rearrangement techniques applied to establish inequalities

Abstract

In this paper we obtain a Hadamard type formula for simple eigenvalues and an analog to the Rayleigh-Faber-Krahn inequality for a class of nonlocal eigenvalue problems. Such class of equations include among others, the classical nonlocal problems with Dirichlet and Neumann conditions. The Hadamard formula is computed allowing domain perturbations given by embeddings of $n$-dimensional Riemannian manifolds (possibly with boundary) of finite volume while the Rayleigh-Faber-Krahn inequality is shown by rearrangement techniques.