Direct integral decomposition for periodic function spaces and application to Bloch waves
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Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago
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2.
Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Andrés Bello s/n, Casilla 447, Chillán
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Received:
01 April 2008
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Primary: 35B27, 46A35, 35Q30; Secondary: 34B27, 47A70.
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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.
Citation: Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves[J]. Networks and Heterogeneous Media, 2008, 3(3): 555-566. doi: 10.3934/nhm.2008.3.555
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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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