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Contributions à l'étude d'espaces de fonctions et d'EDP dans une classe de domaines à frontière fractale auto-similaire

Thibaut Deheuvels 1
1 ACES - Ambient computing and embedded systems
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
Abstract : We study some questions of analysis in view of the modeling of tree-like structures, such as the human lungs. More particularly, we focus on a class of planar ramified domains whose boundary contains a fractal self-similar part. We start by studying some function spaces defined for this class of domains. We first study the Sobolev regularity of the traces on the fractal part of the boundary of functions in some Sobolev spaces of the ramified domains. Then, we study the existence of Sobolev extension operators for the ramified domains we consider. Finally, we compare the notion of self-similar trace on the fractal part of the boundary with more classical definitions of trace. In the last part, we focus on a mixed transmission problem between the ramified domain and the exterior domain. The fractal part of the boundary is the interface of the problem. We propose a numerical approach where we approximate the self-similar interface by a prefractal interface. The proposed strategy is based on a self-similar method for the resolution of the inner problem coupled with an integral method for the resolution of the outer problem.
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Thibaut Deheuvels. Contributions à l'étude d'espaces de fonctions et d'EDP dans une classe de domaines à frontière fractale auto-similaire. Mathématiques générales [math.GM]. Université Rennes 1, 2013. Français. ⟨NNT : 2013REN1S025⟩. ⟨tel-00869946⟩

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