We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter ε > 0. The periodicity cell itself contains a string of holes at a distance O ( ε ) between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as ε → 0. We demonstrate that, for ε small enough, some spectral gaps are open. The position and size of the opened gaps depend on the strip width, the perforation period, and certain integral characteristics of the holes. The asymptotic behavior of the dispersion curves near the band edges is described by means of a ‘fast Floquet variable’ and involves boundary layers in the vicinity of the perforation string of holes. The dependence on the Floquet parameter of the model problem in the periodicity cell requires a serious modification of the standard justification scheme in homogenization of spectral problems. Some open questions and possible generalizations are listed.

In this paper we study an aggregation equation with a general nonlocal flux. We study the local well-posedness and some conditions ensuring global existence. We are also interested in the differences arising when the nonlinearity in the flux changes. Thus, we perform some numerics corresponding to different convexities for the nonlinearity in the equation.