Eigenvalue bounds and spectral asymptotics for fractal Laplacians

Pinasco, Juan Pablo; Scarola, Cristian

doi:10.4171/jfg/71

Cited by 5 publications

(3 citation statements)

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“…This is the main motivation of the present paper. This paper is also a continuation of the work by the authors [8] and by Pinasco and Scarola [32] on estimating the first eigenvalue of Laplacians with respect to fractal measures.…”

Section: Introductionmentioning

confidence: 67%

Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

Deng¹,

Ngai

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Section: Introductionmentioning

confidence: 67%

Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

Deng¹,

Ngai

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…These Laplacians, as well as their generalizations, have been studied extensively in connection with fractal geometry, such as existence of an orthonormal basis of eigenfunctions, spectral dimension and spectral asymptotics, eigenvalues and eigenfunctions, eigenvalue estimates, differential equations, nodal inverse problems, wave equations and wave speed, heat equation and heat kernel estimates, etc. (see [4,7,12,16,[20][21][22][23][24][25]32,35,36,50,51,[53][54][55][56][57]62,63,69] and references therein). We remark that Freiberg and her coauthors defined a class of Laplacians that are more general in that Lebesgue measure is replaced by a more general measure (see, e.g., [21,23]).…”

Section: Introductionmentioning

confidence: 99%

Krein-Feller operators on Riemannian manifolds and Minkowski spaces

Ngai¹,

Ouyang²

2023

Preprint

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For a bounded open set Ω in a complete n-dimensional oriented Riemannian manifold and a positive finite Borel measure µ with support contained in Ω, we define an associated Dirichlet Laplacian ∆ µ by assuming the Poincaré inequality. We obtain sufficient conditions for the Laplacian to have compact resolvent and in this case, we prove the Hodge theorem for functions, which states that there exists an orthonormal basis of L 2 (Ω, µ) consisting of eigenfunctions of ∆ µ , the eigenspaces are finite-dimensional, and the eigenvalues of −∆ µ are real, countable, and increasing to infinity. One of these sufficient conditions is that the L ∞ -dimension dim ∞ (µ) of µ is greater than n − 2. The main tools we use are the Toponogov theorem and the Rauch comparison theorem. We study the condition dim ∞ (µ) > n − 2 for self-similar and self-conformal measures. Results in this paper extend analogous ones by Hu et al. in [J. Funct. Anal. 239 (2006), 542-565], which are established for measures on R n . We also study the d'Alembertian µ = d 2 /dt 2 − ∆ µ on the Minkowski space R 1,n .

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“…These Laplacians are also known as Kreȋn-Feller operators; they were first studied by Kreȋn and Feller in the 1950s [8,9,17,18]. In the fractal setting they have been studied quite extensively, including properties of the Laplacian [12,16], eigenvalues and eigenfunctions [3,6], spectral asymptotics [10,13,28,30,32,34,37], eigenvalue estimates [7,35], wave equation and wave speed [1,4,33], heat equation and heat kernel estimates [14,39], Schrödinger operators [22,31], and inverse problems [11].…”

Section: Introductionmentioning

confidence: 99%

Spectral asymptotics of Laplacians associated with a class of higher-dimensional graph-directed self-similar measures ^*

Ngai

Xie

2021

Nonlinearity

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The spectral dimension of a fractal Laplacian encodes important geometric, analytic, and measure-theoretic information. Unlike standard Laplacians on Euclidean spaces or Riemannian manifolds, the spectral dimension of fractal Laplacians are often non-integral and difficult to compute. The computation is much harder in higher-dimensions. In this paper, we set up a framework for computing the spectral dimension of the Laplacians defined by a class of graph-directed self-similar measures on R d (d ⩾ 2) satisfying the graph open set condition. The main ingredients of this framework include a technique of Naimark and Solomyak and a vector-valued renewal theorem of Lau et al.

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Eigenvalue bounds and spectral asymptotics for fractal Laplacians

Cited by 5 publications

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Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

Krein-Feller operators on Riemannian manifolds and Minkowski spaces

Spectral asymptotics of Laplacians associated with a class of higher-dimensional graph-directed self-similar measures ^*

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Eigenvalue bounds and spectral asymptotics for fractal Laplacians

Cited by 5 publications

References 0 publications

Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

Krein-Feller operators on Riemannian manifolds and Minkowski spaces

Spectral asymptotics of Laplacians associated with a class of higher-dimensional graph-directed self-similar measures *

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Spectral asymptotics of Laplacians associated with a class of higher-dimensional graph-directed self-similar measures ^*