Diffusion and propagation problems in some ramified domains with a fractal boundary

Abstract

Abstract. This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of R 2 with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary. Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous Neumann conditions, the emphasis is placed on transparent boundary conditions, which allow the computation of the solutions in the subdomains obtained by stopping the geometric construction after a finite number of steps. The proposed methods and algorithms will be used numerically in forecoming papers.Mathematics Subject Classification. 28A80, 35J05, 35J25, 65N.

Homogenization of Elastic Media with Gaseous Inclusions

“…Our motivation concerns the mathematical modeling of the respiratory system and this paper belongs to a series relative to this topic ( [1], [2], [26], [17], [18]). Our interest here is to provide the basics to get a model for the behavior of the alveolar region.…”

“…It is easy to see that (31) is satisfied in (LP ) 0 and (LP ) 1 . The condition (31) applied to (LP ) 2 provides a further relation between d 0 and d 1 .…”

Transparent boundary conditions for a class of boundary value problems in some ramified domains with a fractal boundary

“…Le domaine est construit à partir d'un motif de base Y 0 et de deux transformations affines F 1 et F 2 donnés par (1). Pour n 1, soit A n l'ensemble des 2 n applications de {1, .…”

“…C'est précisément ce qui rend possible l'approximation des opérateurs de Dirichlet-Neumann par une formule de récurrence. La méthode est esquissée dans la fin de la note (voir [1] pour l'algorithme complet, ainsi que pour une estimation d'erreur). Le programme proposé a été appliqué numériquement avec succès, voir [2].…”

“…Le programme proposé a été appliqué numériquement avec succès, voir [2]. Les preuves des résultats annoncés dans cette Note sont contenues dans [1].…”