Comparison of Different Definitions of Traces for a Class of Ramified Domains with Self-Similar Fractal Boundaries

Yves Achdou (1) , Thibaut Deheuvels (2) , Nicoletta Tchou (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 IRMAR - Institut de Recherche Mathématique de Rennes

We consider a class of ramified bidimensional domains with a self-similar boundary, which is supplied with the self-similar probability measure. Emphasis is put on the case when the domain is not an epsilon-delta domain as defined by Jones and the fractal is not totally disconnected.We compare two notions of trace on the fractal boundary for functions in some Sobolev space, the classical one ( the strict definition ) and another one proposed in 2007 and heavily relying on self-similarity. We prove that the two traces coincide almost everywhere with respect to the self similar probability measure.

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

Domaines: Equations aux dérivées partielles [math.AP]

Yves Achdou, Thibaut Deheuvels, Nicoletta Tchou. Comparison of Different Definitions of Traces for a Class of Ramified Domains with Self-Similar Fractal Boundaries. Potential Analysis, 2014, 40 (4), pp.345-362. ⟨10.1007/s11118-013-9352-y⟩. ⟨hal-00657954⟩