Affiliation: | 1. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain;2. Institute of Problems of Mechanical Engineering RAS, Saint Petersburg, Russia;3. Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Madrid, Spain Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain;4. Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Santander, Spain |
Abstract: | In this paper, we provide uniform bounds for convergence rates of the low frequencies of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane of height H. The perforations are periodically placed along the ordinate axis at a distance between them, where ε is a parameter that converges toward zero. Another parameter η, the Floquet-parameter, ranges in the interval . The boundary conditions are quasi-periodicity conditions on the lateral sides of the rectangle and Neumann over the rest. We obtain precise bounds for convergence rates which are uniform on both parameters ε and η and strongly depend on H. As a model problem associated with a waveguide, one of the main difficulties in our analysis comes near the nodes of the limit dispersion curves. |