We will study the spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with contains periodic arrangement of inclusions. On the boundary of the waveguide we consider both Neumann and Dirichlet conditions. We will prove that provided the diameter of the inclusion is small enough in the spectrum of Laplacian opens spectral gaps, i.e. frequencies that does not propagate through the waveguide. The existence of the band gaps will verified using the asymptotic analysis of elliptic operators.

We establish that the continuous spectrum of the Neumann problem for the system of elasticity equations occupies the entire closed positive real semiaxis in the case that a three-dimensional body with a sharp-spiked cusp whose cross-section contracts to a point with the velocity O(r 1+γ ), where r is the distance to the vertex of the spike and γ > 1 is the sharpness exponent.

The discrete spectrum of the Dirichlet problem for the Laplace operator on the union of two circular unit cylinders whose axes intersect at the right angle consists of a single eigenvalue. For the threshold value of the spectral parameter, this problem has no bounded solutions. When the angle between the axes reduces, the multiplicity of the discrete spectrum grows unboundedly.

We consider an acoustic waveguide (the Neumann problem for the Helmholtz equation) shaped like a periodic family of identical beads on a thin cylinder rod. Under minor restrictions on the bead and rod geometry, we use asymptotic analysis to establish the opening of spectral gaps and find their geometric characteristics. The main technical difficulties lie in the justification of asymptotic formulas for the eigenvalues of the model problem on the periodicity cell due to its arbitrary shape.

No abstract