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 fromW1,p( ) toW1,p(R2) (1 < p < 1) for a class of tree-shaped domains with a self-similar fractal boundary pre- viously 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 2 [1,1]. 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.

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. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2016, XV (special), pp.209-247. ⟨10.2422/2036-2145.201307_008⟩. ⟨hal-00659241v2⟩