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

Abstract

International audienceWe 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

Sobolev extension property for tree-shaped domains with self-contacting fractal boundary

“…As was seen in Remark 5.1, it is proved a posteriori in [2] that the trace operator 1 coincides with the trace operator introduced by Jonsson and Wallin in [17] µalmost everywhere. Therefore, if is a W 1, p -extension domain for some p > p ?…”

“…A consequence of the main result of this paper is that these two definitions of trace on 0 1 in fact coincide (almost everywhere) on the set 0 1 ; this is proved in [2].…”

“…Another choice could have been to work with the Whitney extension operator of (5.1), but we then could not have exploited the self-similar properties of the geometry as is done to prove Theorem B. In [2], Theorems A and B below are key ingredients in the proof that u| 0 1 =`1(u) µ-almost everywhere for all p > 1 and u 2 W 1, p ().…”

Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries

“…Assumption 2 does not allow the boundary to collapse into an infinitely thing structures, as for instance happens in the fractal threes. Actually it is the reason why the fractal threes [1] are not (ε, ∞)-domains.…”

“…In Section 5 we continue the generalization and show the compactness of the trace operator considered this time as an operator mapping not on its image, but in L p (∂Ω). Section 6 gives an example of the application of obtained theorems by showing the well-posedness of the Poisson problem (1) on the H 1 -Sobolev admissible domains with a standard notation W 1 2 (Ω) = H 1 (Ω).…”