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

Abstract

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u)=−∫RdK(x,y)(u(y)−u(x))dy. Here we consider a kernel K(x,y)=ψ(y−a(x))+ψ(x−a(y)) where ψ is a bounded, nonnegative function supported in the unit ball and a means a diffeomorphism on Rd. 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 ψ. Indeed, in the linear case a(x)=Ax we obtain an explicit expression for the first eigenvalue in the whole Rd and it is positive when 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 behavior of the principal eigenvalue of the nonlocal Dirichlet problem in the ball BR and prove that it converges to the first eigenvalue in the whole space as R→∞.