Trace Results on Domains with Self-Similar Fractal Boundaries

Yves Achdou (1) , Nicoletta Tchou (2)

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

This work deals with trace theorems for a family of ramified domains $\Omega$ with a self-similar fractal boundary $\Gamma^\infty$. The fractal boundary $\Gamma^\infty$ is supplied with a probability measure $\mu$ called the self-similar measure. Emphasis is put on the case when the domain is not a $\epsilon-\delta$ domain and the fractal is not post-critically finite, for which classical results cannot be used. It is proven that the trace of a square integrable function belongs to $L^p_\mu$ for all real numbers $p\ge 1$. A counterexample shows that the trace of a function in $H^1(\Omega)$ may not belong to $BMO(\mu)$ (and therefore may not belong to $L^\infty_\mu$). Finally, it is proven that the traces of the functions in $H^1(\Omega)$ belong to $H^s(\Gamma^\infty)$ for all real numbers $s$ such that $0\le s d_H/4$ are supplied. \\ There is an important contrast with the case when $\Gamma^\infty$ is post-critically finite, for which the square integrable functions have their traces in $H^s(\Gamma^\infty)$ for all $s$ such that $0\le s

Mots clés: self-similar domain; fractal boundary; partial differential equations

Domaines: Equations aux dérivées partielles [math.AP]

Yves Achdou, Nicoletta Tchou. Trace Results on Domains with Self-Similar Fractal Boundaries. Journal de Mathématiques Pures et Appliquées, 2008, 89 (6), pp.596-623. ⟨10.1016/j.matpur.2008.02.008⟩. ⟨hal-00202658⟩