Sobolev Extension Property for Tree-shaped Domains with Self-contacting Fractal Boundary

In this paper, we investigate the existence of extension operators fromW (Ω) toW (R) (1 < p < ∞) for a class of tree-shaped domains Ω with a self-similar fractal boundary previously studied by Mandelbrot and Frame. When the fractal boundary has no self-contact, the results of Jones imply that there exist such extension operators for all p ∈ [1,∞]. In the case when the fractal boundary self-intersects, this result does not hold. Here, we prove however that extension operators exist for p < p where p depends only on the dimension of the self-intersection of the boundary. The construction of these operators mainly relies on the self-similar properties of the domains.