Boundary Value Problems in Ramified Domains with Fractal Boundaries

This work deals with some Poisson problems in a self-similar ramified domain of ℝ² with a fractal boundary (see Figure 1). We consider generalized Neumann condition on the fractal boundary. The first goal is to give a rigorous functional setting. The second goal is to propose a strategy for computing the solutions in simple subdomains obtained by stopping the construction after a finite number of steps. When the Neumann data belongs to the Haar basis associated to a dyadic decomposition of the fractal boundary, we show that the solution can be found by solving a sequence of boundary value problems in an elementary cell, with nonhomogeneous and nonlocal boundary conditions. For a general Neumann data g, the idea is to expand g on the Haar basis and use the linearity of the problem for deriving an expansion of the solution.