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

Carmen Calvo-Jurado; Juan Casado-Díaz; Manuel Luna-Laynez

doi:10.3934/proc.2013.2013.85

Article Contents

2013, Volume 2013: 85-94. Doi: 10.3934/proc.2013.2013.85 open access

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

1.
Dpto. de Matemáticas. Escuela Politécnica, Avenida de la Universidad s/n, 10003 Cáceres
2.
Dpto. de Ecuaciones Diferenciales y Análisis Numérico., Fac. de Matemáticas. C. Tarfia s/n., 41012 Sevilla

Received: August 31, 2012

Published: October 2013

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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.

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Mathematics Subject Classification: Primary: 35B27; Secondary: 35R60.

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