Partial Differential Equations
Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths
Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 133-138.

We study the asymptotic behavior of the solution of a diffusion problem posed in the union of a cylinder of small diameter and fixed length with another cylinder with much smaller diameter and length. The Dirichlet condition is assumed to hold at both extremities of this domain. Depending on the relative size of the parameters, we show that the boundary condition of the one-dimensional limit problem is a Dirichlet, Fourier or Neumann condition. We also prove a corrector result for every case.

Nous étudions le comportement asymptotique de la solution d'un problème de diffusion posé sur l'union d'un cylindre de petit diamètre et de longueur fixe et d'un autre cylindre de longueur et de diamètre beaucoup plus petits. La condition de Dirichlet est imposée aux deux extrémités. Nous démontrons que selon les valeurs relatives des paramètres, la condition au bord du problème unidimensionnel limite est une condition de Dirichlet, de Fourier ou de Neumann. Nous démontrons aussi dans chaque cas un résultat de correcteur.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.10.033
Casado-Díaz, Juan 1; Luna-Laynez, Manuel 1; Murat, François 2

1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75252 Paris cedex 05, France
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Casado-Díaz, Juan; Luna-Laynez, Manuel; Murat, François. Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths. Comptes Rendus. Mathématique, Volume 338 (2004) no. 2, pp. 133-138. doi : 10.1016/j.crma.2003.10.033. http://www.numdam.org/articles/10.1016/j.crma.2003.10.033/

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