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

✍ Scribed by António M. Caetano; Sofia Lopes

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 327 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200910214

✦ Synopsis

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 bounded

where (-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.