Trace results on domains with self-similar fractal boundaries |
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Authors: | Yves Achdou Nicoletta Tchou |
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Institution: | aUFR Mathématiques, Université Paris Diderot, Case 7012, 75251 Paris Cedex 05, France;bLaboratoire Jacques-Louis Lions, Université Paris 6, 75252 Paris Cedex 05, France;cIRMAR, Université de Rennes 1, Rennes, France |
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Abstract: | This work deals with trace theorems for a family of ramified bidimensional domains Ω with a self-similar fractal boundary Γ∞. The fractal boundary Γ∞ is supplied with a probability measure μ called the self-similar measure. Emphasis is put on the case when the domain is not a −δ domain and the fractal is not post-critically finite, for which classical results cannot be used. It is proven that the trace of a function in H1(Ω) belongs to for all real numbers p1. A counterexample shows that the trace of a function in H1(Ω) may not belong to BMO(μ) (and therefore may not belong to ). Finally, it is proven that the traces of the functions in H1(Ω) belong to Hs(Γ∞) for all real numbers s such that 0s<dH/4, where dH is the Hausdorff dimension of Γ∞. Examples of functions whose traces do not belong to Hs(Γ∞) for all s>dH/4 are supplied.There is an important contrast with the case when Γ∞ is post-critically finite, for which the functions in H1(Ω) have their traces in Hs(Γ∞) for all s such that 0s<dH/2. |
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Keywords: | Self-similar domain Fractal boundary Sobolev spaces Trace theorems |
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