The dual of Besov spaces on fractals

Jonsson, Alf; Wallin, Hans

doi:10.4064/sm-112-3-285-300

Cited by 52 publications

(41 citation statements)

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“…Here J stands for jumps, since the considered functions may jump at some points of K. If the fractal set K is totally disconnected, then these spaces coincide with the Lipschitz spaces Lip(α, p, q; m; K) also introduced in [10]. The latter are a generalization of the more classical spaces Lip(α, p, q; K) introduced in [11] since Lip(α, p, q; [12]. Since this is sufficient for what follows, we will focus on the case when K = Γ ∞ and p = q.…”

Section: Lipschitz Functions With Jumps On γ ∞mentioning

confidence: 99%

Trace Theorems for a Class of Ramified Domains with Self-Similar Fractal Boundaries

Achdou¹,

Tchou²

2010

SIAM J. Math. Anal.

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This work deals with trace theorems for a class 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 as defined by Jones and the fractal set is not totally disconnected. In this case, the classical trace results cannot be used. Here, the Lipschitz spaces with jumps recently introduced by Jonsson play a crucial role. Indeed, it is proved in particular that if the Hausdorff dimension d of Γ ∞ is not smaller than one, then the space of the traces ofThe proof is elementary; a main step is a strengthened trace inequality in the norm L q µ (Γ ∞ ).

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Section: Lipschitz Functions With Jumps On γ ∞mentioning

confidence: 99%

Trace Theorems for a Class of Ramified Domains with Self-Similar Fractal Boundaries

Achdou¹,

Tchou²

2010

SIAM J. Math. Anal.

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“…There are many equivalent definitions [23,31] of Besov spaces. To give one of them, we introduce [21,23] a net N with mesh 2 −ν , ν ∈ N , i.e.…”

Section: A Definitions Of Besov Spaces On Fractalsmentioning

confidence: 99%

“…We will see that the existence and uniqueness results of a weak solution of the problem (1)-(4) hold for a class of bounded (ǫ, δ) -domains [20,21,22] Ω + such that ∂Ω is a d -set [21]: [21,22,23]) Let Γ be a closed subset of R n and 0 < d ≤ n . A positive Borel measure m d with support Γ is called a d -measure of Γ if, for some positive constants c 1 , c 2 > 0 ,…”

Section: Extension Tomentioning

confidence: 99%

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Short-time heat diffusion in compact domains with discontinuous transmission boundary conditions

Bardos

Rozanova-Pierrat

2015

Math. Models Methods Appl. Sci.

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We consider a heat problem with discontinuous diffusion coefficients and discontinuous transmission boundary conditions with a resistance coefficient. For all compact (ǫ, δ) -domains Ω ⊂ R n with a d -set boundary (for instance, a self-similar fractal), we find the first term of the small-time asymptotic expansion of the heat content in the complement of Ω , and also the second-order term in the case of a regular boundary. The asymptotic expansion is different for the cases of finite and infinite resistance of the boundary. The derived formulas relate the heat content to the volume of the interior Minkowski sausage and present a mathematical justification to the de Gennes' approach. The accuracy of the analytical results is illustrated by solving the heat problem on prefractal domains by a finite elements method.

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“…Our approach is intimately linked with function spaces on R n and on F. Hence, the second aim of this paper is to introduce some function spaces of Besov type on F. In that sense this note might be considered as a direct continuation of our paper [13]. On the other hand function spaces on (closed) subsets of R ~ have been studied extensively by A. Jonsson and H. Wallin, see [5], [6], [7], [8] and [15]. Our approach is closely related to this work and should also be seen in the context of the theory developed there.…”

Section: Introductionmentioning

confidence: 99%