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The homogenization of the heat equation with mixed conditions on randomly subsets of the boundary

Pages: 85 - 94, Issue special, November 2013

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Carmen Calvo-Jurado - Dpto. de Matemáticas. Escuela Politécnica, Avenida de la Universidad s/n, 10003 Cáceres, Spain (email)
Juan Casado-Díaz - Dpto. de Ecuaciones Diferenciales y Análisis Numérico., Fac. de Matemáticas. C. Tarfia s/n., 41012 Sevilla, Spain (email)
Manuel Luna-Laynez - Dpto. de Ecuaciones Diferenciales y Análisis Numérico., Fac. de Matemáticas. C. Tarfia s/n., 41012 Sevilla, Spain (email)

Abstract: We consider a domain in $\mathbb{R}^N$, $N\geq 3$, such that a portion of its boundary is plane. In this portion we fix a sequence $K_\epsilon$ of small subsets randomly distributed in such way that the distance between them is of order $\epsilon$ and their diameters are of order $\epsilon^\frac{N-1}{N-2}$. We study the asymptotic behavior of the heat equation with Dirichlet conditions on $K_\epsilon$ and Neumann conditions on the rest of the boundary. We prove the convergence to a limit problem with a Fourier-Robin boundary condition which has the physical interest of being deterministic.

Keywords:  Homogenization, linear problems, random perforated domains.
Mathematics Subject Classification:  Primary: 35B27; Secondary: 35R60.

Received: September 2012; Published: November 2013.

 References