On Band-Gap Structure of Spectrum in Network Double-Porosity Models

Abstract

We study spectral properties of periodic divergent-type elliptic operator $A_{\varepsilon }$ with high contrast coefficients on $\varepsilon$-periodic thin network $F_{\varepsilon }$, which is asymptotically singular and can be obtained from 1-periodic graph F by means of fattening and contraction. The network $F_{\varepsilon }$ is divided into stiff and soft parts, also 1-periodic, where the coefficients of $A_{\varepsilon }$ are of order 1 and $\varepsilon ^{2}$, respectively; the stiff part is connected and the soft part is dispersive. We prove that the spectrum of the operator $A_{\varepsilon }$ has the band-gap structure and show the existence of non-degenerate spectral bands and open gaps, the number of which grows to infinity as $\varepsilon \rightarrow 0$. We establish connection between the endpoints of gaps and eigenvalues of two operators defined on the cell of periodicity. The first is the Laplace–Dirichlet operator on the “soft” part of the graph F within the unit cell and the second is its electrostatic extension onto the whole cell. Moreover, the band-gap structure of the spectrum can be described asymptotically exactly on each finite interval under additional geometric condition which is the “smallness” of the soft phase with respect to the stiff phase.