Summary: Diffusion and Propagation Problems in Some Ramified Domains
with a Fractal Boundary
, Christophe Sabot
, Nicoletta Tchou
February 19, 2007
This paper is devoted to some elliptic boundary value problems in a self-similar ramified
domain of R2
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.