Direct integral decomposition for periodic function spaces and application to Bloch waves
Carlos Conca, ,Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Luis Friz, Jaime H. Ortega, ,Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Andrés Bello s/n, Casilla 447, Chillán · Networks and Heterogeneous Media · 2008
In this paper, we study a direct integral decompositionfor the spaces $L^2(O)$ and $H^1(O)$ based on $(\xi,Y^*)-$periodicfunctions. Using this decomposition we can write the Green'soperator (associated to the classical Stokes system in fluid mechanics)in terms of a family of self-adjoint compact operators which depend on theparameter $\xi$. As a consequence, we obtain theso-called Bloch waves associated to the Stokes system in the caseof a periodic perforated domain.