Sobolev extension property for tree-shaped domains with self-contacting fractal boundary

Thibaut Deheuvels (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper, we investigate the existence of extension operators from $W^{1,p}(\Omega)$ to $W^{1,p}(\R^2)$ (p≥1) for a class of tree-shaped domains $\Omega$ with a self-similar fractal boundary previously studied by Mandelbrot and Frame. Such a geometry can be seen as a bidimensional modelization of the bronchial tree. When the fractal boundary has no self-contact, Jones proved 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

Mots clés: Self-similar domain; Fractal boundary; Sobolev extension domain; Traces; Partial differential equations

Domaines: Equations aux dérivées partielles [math.AP]

Thibaut Deheuvels. Sobolev extension property for tree-shaped domains with self-contacting fractal boundary. 2012. ⟨hal-00659241v1⟩