| Issue |
ESAIM: M2AN
Volume 40, Number 4, July-August 2006
|
|
|---|---|---|
| Page(s) | 623 - 652 | |
| DOI | https://doi.org/10.1051/m2an:2006027 | |
| Published online | 15 November 2006 | |
Diffusion and propagation problems in some ramified domains with a fractal boundary
1
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. achdou@math.jussieu.fr
2
CNRS, UMPA, UMR 5669,
46, Allée d'Italie,
69364 Lyon Cedex 07,
France. csabot@umpa.ens-lyon.fr
3
IRMAR, Université de Rennes 1, Rennes, France. nicoletta.tchou@univ-rennes1.fr
Received:
26
September
2005
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
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