Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space

TL;DR

This paper establishes bounds for the first eigenvalue of a nonlocal diffusion operator with a specific kernel, providing explicit formulas in the linear case and analyzing the eigenvalue's implications for solution decay.

Contribution

It introduces bounds and explicit formulas for the first eigenvalue of a nonlocal diffusion operator with a particular kernel, extending understanding in the whole space setting.

Findings

Explicit eigenvalue formula for linear case when det(A) ≠ 1

Eigenvalue positivity implies exponential decay of solutions

Eigenvalues in bounded domains converge to the whole space eigenvalue as radius increases

Abstract

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $ T(u) = - \int_{\rr^d} K(x,y) (u(y)-u(x)) \, dy$. Here we consider a kernel $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$ where $\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ means a diffeomorphism on $\rr^d$. A simple example being a linear function $a(x)= Ax$. The upper and lower bounds that we obtain are given in terms of the Jacobian of $a$ and the integral of $\psi$. Indeed, in the linear case $a(x) = Ax$ we obtain an explicit expression for the first eigenvalue in the whole $\rr^d$ and it is positive when the the determinant of the matrix $A$ is different from one. As an application of our results, we observe that, when the first eigenvalue is positive, there is an exponential decay for the solutions to the associated evolution problem. As a tool to obtain the result, we also study the behaviour of the principal eigenvalue of the nonlocal Dirichlet problem in the ball $B_R$ and prove that it converges to the first eigenvalue in the whole space as $R\to \infty$.