Homogenization of nonlocal spectral problems

We consider a spectral problem for convolution-type operators in environments with locally periodic microstructure and study the asymptotic behavior of the bottom of the spectrum. We show that the bottom point of the spectrum converges as the microstructure period tends to zero, and identify the limit in terms of an additive eigenvalue problem for effective Hamilton–Jacobi equation. In the periodic case, we establish a more accurate two-term asymptotic formula.