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

Carlos Conca; Luis Friz; Jaime H. Ortega

doi:10.3934/nhm.2008.3.555

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2008, Volume 3, Issue 3: 555-566. Doi: 10.3934/nhm.2008.3.555

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

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

Received: April 2008

Published: September 2008

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B27, 46A35, 35Q30; Secondary: 34B27, 47A70.

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