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

Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary

“…The first goal is to give a rigorous functional setting allowing the study of variational formulations. In contrast with [2], trace results in Sobolev spaces are needed; to our knowledge, these results are new. The second goal is to propose a method for computing the solutions in simple subdomains of Ω 0 .…”

“…This Note deals with some boundary value problems in a self-similar ramified domain of R 2 with a fractal boundary: the domain Ω 0 whose construction is presented in Section 2 below, (see also the figure in [2]), has the following important property: there exists a straight line Γ ∞ (see (4) below), such that (i) Γ ∞ ⊂ ∂Ω 0 , (ii) ∀x ∈ Γ ∞ , ∀r > 0, Ω 0 ∩ B(x, r) is not an ( , δ) domain, see e.g. [3] or [4], where B(x, r) is the two-dimensional ball with center x and radius r.…”

“…The Note [2] was devoted to Poisson problems with a generalized homogeneous Neumann condition on Γ ∞ . Both Laplace and Helmholtz equation were considered.…”

“…The second goal is to propose a method for computing the solutions in simple subdomains of Ω 0 . We use the Dirichlet-Neumann operator T introduced in [2]. When the Neumann data belongs to the Haar basis, we propose new boundary value problems in simple subdomains of Ω 0 , with nonhomogeneous and nonlocal boundary conditions involving T .…”

“…affine M σ (F 1 , F 2 ) par (2). Le domaine Ω 0 est construit dans (3) en assemblant les domaines Y 0 et M σ (F 1 , F 2 )(Y 0 ),…”