Dirichlet Forms on Fractal Subsets of the Real Line

Freiberg, Uta

doi:10.14321/realanalexch.30.2.0589

Cited by 22 publications

(16 citation statements)

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“…It is known (see [15,Theorem 4.1]) that (E, F) defines a Dirichlet form on L 2 ([0, 1], µ). Hence, there exists an associated non-negative, self-adjoint operator…”

Section: Preliminariesmentioning

confidence: 99%

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

Ehnes

Hambly

2020

Preprint

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We consider the heat equation defined by a generalized measure theoretic Laplacian on [0, 1]. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure µ, where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding heat equations. This provides, for example, an interpretation for the mathematical model of heat diffusion on a bar with gaps in that the solution to the corresponding heat equation behaves approximately like the heat flow on a bar with sufficiently small mass on these gaps.

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“…It is known (see [15,Theorem 4.1]) that (E, F) defines a Dirichlet form on L 2 ([0, 1], µ). Hence, there exists an associated non-negative, self-adjoint operator…”

Section: Preliminariesmentioning

confidence: 99%

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

Ehnes

Hambly

2020

Preprint

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“…(k α T k ) k ≤ lim inf t→∞ log N (I) D (e(1 + log t) α t) log t a.s. Since lim t→∞ log e(1 + log t) α t log t = 1, lim t→∞ e(1 + log t) α t = ∞, With the inequality N µ D (x) ≤ N µ N (x) ≤ N µ D (x) + 2, x ≥ 0,for arbitrary finite atomless Borel measure µ (see[6, Proposition 5.5]), we also receive lim…”

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confidence: 93%

Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures

Minorics

2019

Forum Mathematicum

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AbstractWe study the limiting behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second-order differential operators {\frac{\mathop{}\!d}{\mathop{}\!d\mu}\frac{\mathop{}\!d}{\mathop{}\!dx}}, where μ is a finite atomless Borel measure on some compact interval {[a,b]}. Therefore, we firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we make a proposition about the convergence behavior for so-called random V-variable Cantor measures.

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“…Note that the function g (y) does not always exist and is not unique for a given g(y). If g ∈ L 2 (F, m), then it solves both of these problems [6,19,[46][47][48][49]. F α -Calculus approach: F α -Calculus (F α -C) is a simple, constructive, and algorithmic approach to the analysis of fractals [20,21].…”

Section: Introductionmentioning

confidence: 99%

Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

Golmankhaneh

Tunç

2019

Fractal Fract

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In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether’s Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results.

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Dirichlet Forms on Fractal Subsets of the Real Line

Cited by 22 publications

References 0 publications

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures

Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

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