Operator estimates for the crushed ice problem

Khrabustovskyi, Andrii; Post, Olaf

doi:10.3233/asy-181480

Cited by 18 publications

(21 citation statements)

References 33 publications

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“…Moreover, although the concept of quasi-unitary equivalence is stronger than e.g. Mosco convergence, it seems to be not very difficult to show the conditions in Definition 2.1 in applications such at graphs approximating self-similar fractals or in the above homogenisation setting of [KP18].…”

Section: Quasi-unitary Equivalencementioning

confidence: 99%

Approximation of Fractals by Discrete Graphs: Norm Resolvent and Spectral Convergence

Post

Simmer

2018

Integr. Equ. Oper. Theory

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We show a norm convergence result for the Laplacian on a class of pcf self-similar fractals with arbitrary Borel regular probability measure which can be approximated by a sequence of finite-dimensional weighted graph Laplacians.As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.Our analysis works for a class of pcf self-similar sets (see Definition 3.1) which we call here fractals approximable by finite weighted graphs (see Definition 3.4). For such a fractal K there is a sequence (G m ) m∈N 0 of nested graphs G m = (V m , E m ) (i.e., V m ⊂ V m+1 ⊂ K) and conductances (i.e., edge weights) c e,m > 0 of the edges e ∈ E m , such that V * = m V m is dense in K, (see e.g. Figure 1 for the pentagasket with all five fixed points as boundary vertices) together with a compatible and self-similar sequence of graph energies (E m ) m (see Definitions 3.2 and 3.3) given byThe compatibility roughly means that E m (ϕ) agrees with the energy of E m+1 (h) where h : V m+1 −→ C is the harmonic extension of ϕ : V m −→ C. The harmonic extension h has the property that it minimises the energy E m (u) among all extensions u : V m+1 −→ C with u Vm = ϕ, see (3.2).

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Section: Quasi-unitary Equivalencementioning

confidence: 99%

Approximation of Fractals by Discrete Graphs: Norm Resolvent and Spectral Convergence

Post

Simmer

2018

Integr. Equ. Oper. Theory

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“…First, the author believes that the norm convergence result generalizes to unbounded domains (that is, when the limit domain is an unbounded interval). A suitable modification of the argument in [CDR17] or [KP17] seems like a reasonable approach.…”

Section: Discussionmentioning

confidence: 99%

“…[Zhi00,Pas06] for perforated domains of fixed size with Neumann boundary conditions, [MS10] for perforated domains with periodic boundary conditions, and [BCD16] for domains perforated along a curve. Advances towards operator norm and spectral convergence in perforated domains have been made in [Pas06,BCD16,CDR17,KP17]). A result by Cioranescu and Murat gives a positive answer to the question of convergence of solutions in the case where the size of \Omega \varepsi remains constant but the holes shrink and concentrate.…”

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confidence: 99%

A Strange Vertex Condition Coming from Nowhere

Rösler¹

2021

SIAM J. Math. Anal.

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We prove norm-resolvent and spectral convergence in L 2 of solutions to the Neumann Poisson problem - \Delta u\varepsi = f on a domain \Omega \varepsi perforated by Dirichlet holes and shrinking to a 1dimensional interval. The limit u satisfies an equation of the type - u \prime \prime + \mu u = f on the interval (0, 1), where \mu is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighborhood and the vertex neighborhood is chosen correctly, the constant \mu will appear in the vertex condition of the limit problem. In particular, this implies that the spectrum of the resulting quantum graph is altered in a controlled way by the perforation.

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“…From the abstract theory of quasi‐unitary equivalence of energy forms we deduce the following (for more consequences we also refer to [29, Sec. 3]): Theorem Assume that

scriptE

and

\overset{E}{true}

are δ‐quasi‐unitarily equivalent, then the associated operators

Δ \geq 0

and

\tilde{normalΔ} \geq 0

fulfil the following:

\begin{matrix} true \\ right true∥ η false(normalΔ false) - J^{*} η true(\overset{Δ}{true} true) J true∥ & left \leq C_{η} δ, & right true∥ η true(\overset{Δ}{true} true) - J η false(normalΔ false) J^{*} true∥ & left \leq C_{η}^{'} δ, \\ right prefixdist (() \frac{1}{1 + σ_{} false(normalΔ false)}, \frac{1}{1 + σ_{} false(\overset{Δ}{true} false)}) & left \leq ε false(δ false), & right |λ_{k} false(normalΔ false) - λ_{k} (\tilde{normalΔ})| & left \leq C_{k} δ, \end{matrix}

where η is a suitable function holomorphic in a neighbourhood of

σ_{} false(normalΔ false)

, e.g.,

η_{z} false(λ false) = {(λ - z)}^{- p}

(resolvent powers in z ),

η_{t} false(λ false) = {normale}^{- t λ}

(heat operator) or

η = 1_{I}

with

\partial I \cap σ_{} false(normalΔ

…”

Section: Introductionmentioning

confidence: 99%

Graph‐like spaces approximated by discrete graphs and applications

Post

Simmer

2021

Mathematische Nachrichten

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We define a distance between energy forms on a graph‐like metric measure space and on a suitable discrete weighted graph using the concept of quasi‐unitary equivalence. We apply this result to metric graphs, graph‐like manifolds (e.g. a small neighbourhood of an embedded metric graph) or pcf self‐similar fractals as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary), and express the distance of the associated energy forms in terms of geometric quantities. In particular, we show that there is a sequence of domains converging to a pcf self‐similar fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. As a consequence, the spectra and suitable functions of the associated Laplacians converge, the latter in operator norm.

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Operator estimates for the crushed ice problem

Cited by 18 publications

References 33 publications

Approximation of Fractals by Discrete Graphs: Norm Resolvent and Spectral Convergence

Approximation of Fractals by Discrete Graphs: Norm Resolvent and Spectral Convergence

A Strange Vertex Condition Coming from Nowhere

Graph‐like spaces approximated by discrete graphs and applications

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