Diffusion and propagation problems in some ramified domains with a fractal boundary
UFR Mathématiques, Université Paris 7, Case 7012,
75251 Paris Cedex 05, France and Laboratoire Jacques-Louis Lions, Université Paris 6, 75252 Paris Cedex 05, France. firstname.lastname@example.org
2 CNRS, UMPA, UMR 5669, 46, Allée d'Italie, 69364 Lyon Cedex 07, France. email@example.com
3 IRMAR, Université de Rennes 1, Rennes, France. firstname.lastname@example.org
Revised: 1 March 2006
This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of 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
Key words: Domains with fractal boundaries / Helmholtz equation / Neumann boundary conditions / transparent boundary conditions.
© EDP Sciences, SMAI, 2006