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September  2008, 3(3): 555-566. doi: 10.3934/nhm.2008.3.555

Direct integral decomposition for periodic function spaces and application to Bloch waves

Carlos Conca 1, , Luis Friz 2, and  Jaime H. Ortega 2,

1. 

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago
2. 

Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Andrés Bello s/n, Casilla 447, Chillán, Chile

Received  April 2008 Published  June 2008

In this paper, we study a direct integral decomposition for the spaces $L^2(O)$ and $H^1(O)$ based on $(\xi,Y^*)-$periodic functions. Using this decomposition we can write the Green's operator (associated to the classical Stokes system in fluid mechanics) in terms of a family of self-adjoint compact operators which depend on the parameter $\xi$. As a consequence, we obtain the so-called Bloch waves associated to the Stokes system in the case of a periodic perforated domain.
Keywords: Bloch waves, Direct integral, periodic function.
Mathematics Subject Classification: Primary: 35B27, 46A35, 35Q30; Secondary: 34B27, 47A7.
Citation: Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks and Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555
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