Spectral theory for the fractal Laplacian in the context of h-sets

Abstract

An h-set is a nonempty compact subset of the Euclidean n-space which supports a finite Radon measure for which the measure of balls centered on the subset is essentially given by the image of their radius by a suitable function h. In most cases of interest such a subset has Lebesgue measure zero and has a fractal structure.Let Ω be a boundedwhere (−Δ) −1 is the inverse of the Dirichlet Laplacian in Ω and tr Γ is, say, a trace type operator. The operator B, acting in convenient function spaces in Ω, is studied. Estimations for the eigenvalues of B are presented, and generally shown to be dependent on h, and the smoothness of the associated eigenfunctions is discussed. Some results on Besov spaces of generalised smoothness on R n and on domains which were obtained in the course of this work are also presented, namely pointwise multipliers, the existence of a universal extension operator, interpolation with function parameter and mapping properties of the Dirichlet Laplacian.

The fractal Dirichlet Laplacian

“…145, 194-196 The proof of the next theorem, which extends the results in [28,Theorem 19.7] and [20, Theorem 4.1.7], can be found in [6]. …”

“…Remark 4.5 In [6] there are some more results on the eigenfunctions associated to the operator B, which are not presented here. It is presented a class of Besov spaces of generalised smoothness where the eigenfunctions do and do not belong to.…”

“…If is a domain, then the spaces W m p ( ) and H s p ( ) are defined by restriction of the corresponding spaces on R n . For more details we refer to [6], for example. In this paper we will work with the spaces W 1 2 ( ) = H 1 2 ( ) and H −1 2 ( ).…”

“…The case where is a (d, ψ)-set was studied by Edmunds and Triebel in [9] and by Moura in [19]. The case where is an h-set was studied in [6]. As it was mentioned in these works, in the case n = 2 the operator B has physical relevance: it describes the vibration of a drum where the whole mass is distributed on .…”

Traces of Besov spaces on fractal $h$-sets and dichotomy results

“…Dealing with spac! es on h-sets we refer to [CaL2,CaL1,KZ,Lo], and, probably closest to our approach here, [Tr6,Chapter 8]. There it turns out that one first needs a sound knowledge about the existence and quality of the corresponding trace spaces.…”