On eigenvalue problems for Laplacians on P.C.F. self-similar sets

Shima, T.

doi:10.1007/bf03167295

Cited by 78 publications

(88 citation statements)

References 7 publications

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“…There are, of course, many more examples in which the number of terminal points is finite, such as the Sierpinski gaskets based on higher dimensional simplices. One would also want the spectral decimation method of [Sh2] or [MT] to hold.…”

Section: Fractafolds Based On the Sierpinski Gasket 4021mentioning

confidence: 99%

“…In fact, the matching conditions provide necessary and sufficient conditions for gluing together local solutions ∆u i = f i on C i to obtain a global solution ∆u = f (here we assume that the glued functions u and f are continuous). The spectrum of the Laplacian on SG (with Dirichlet or Neumann boundary conditions) was described exactly by Fukushima and Shima [FS], [Sh1] (see also [DSV], [MT], [T] and [GRS] for further elaborations) based on the method of spectral decimation [Sh2]. Our goal is to give an analogous description for fractafolds.…”

Section: Spectrum Of the Laplacianmentioning

confidence: 99%

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Fractafolds based on the Sierpinski gasket and their spectra

Strichartz

2003

Trans. Amer. Math. Soc.

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Abstract. We introduce the notion of "fractafold", which is to a fractal what a manifold is to a Euclidean half-space. We specialize to the case when the fractal is the Sierpinski gasket SG. We show that each such compact fractafold can be given by a cellular construction based on a finite cell graph G, which is 3-regular in the case that the fractafold has no boundary. We show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph, using the spectral decimation method of Fukushima and Shima. This enables us to obtain isospectral pairs of nonhomeomorphic fractafolds. We also show that although SG is topologically rigid, there are fractafolds based on SG that are not topologically rigid.

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Section: Fractafolds Based On the Sierpinski Gasket 4021mentioning

confidence: 99%

Section: Spectrum Of the Laplacianmentioning

confidence: 99%

Fractafolds based on the Sierpinski gasket and their spectra

Strichartz

2003

Trans. Amer. Math. Soc.

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“…Then −∆ on dom L 2 ∆ becomes a positive selfadjoint operator under Dirichlet boundary conditions with compact inverse (or under Neumann boundary conditions with compact resolvant), and so has a discrete spectrum, with eigenfunctions belonging to domD. In fact the nature of the eigenvalues and eigenfunctions is known explicity via the method of spectral decimation of Fukushima and Shima ([4], [15], [16]). For functions in domD there is also a pointwise formula for ∆u as a limit of a difference quotient…”

Section: E(u)mentioning

confidence: 99%

Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket

Azzam

Hall

Strichartz

2007

Trans. Amer. Math. Soc.

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Abstract. On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy E ϕ and conformal Laplacian ∆ ϕ for a given conformal factor ϕ, based on the corresponding notions in Riemannian geometry in dimension n = 2. We derive a differential equation that describes the dependence of the effective resistances of E ϕ on ϕ. We show that the spectrum of ∆ ϕ (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function ϕ does not vary too much). We illustrate these results with numerical approximations. We give a linear extension algorithm to compute the energy measures of harmonic functions (with respect to the standard energy), and as an application we show how to compute the L p dimensions of these measures for integer values of p ≥ 2. We derive analogous linear extension algorithms for energy measures on related fractals.

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“…The Laplacian on the Vicsek set has been studied extensively in [Barlow 1998;Malozemov and Teplyaev 2003;Metz 1993;Shima 1996], both analytically and probabilistically. Various problems have been studied for this fractal, including topological rigidity [Strichartz 2006], the uniqueness of Brownian motion [Metz 1993], etc.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we study the spectrum of the Laplacian on a special case of this infinite family of fractals, the symmetric n-branch Vicsek set ᐂ n . All eigenvalues of the Laplacian can be obtained through an iterative process called spectral decimation, which was introduced by Shima [1996]. It turns out that the spectral decimation function is associated with the Chebyshev polynomials.…”

Section: Introductionmentioning

confidence: 99%