Gap opening around a given point of the spectrum of a cylindrical waveguide by means of gentle periodic perturbation of walls

We discuss one of the main questions in band-gap engineering, namely by an asymptotic analysis it is proven that any given point of a certain interval in the spectrum of a cylindrical waveguide can be surrounded with a spectral gap by means of a periodical perturbation of the walls. Both the Dirichlet and Neumann boundary conditions for the Laplace operator are considered in planar and multi-dimensional waveguides.