Spectral Asymptotics of Some One-Dimensional Fractal Laplacians

Sze-Man Ngai

Spectral Asymptotics of Some One-Dimensional Fractal Laplacians

Mathematical Sciences

Research output: Contribution to conference › Presentation

Abstract

<p> The spectral dimension of the Laplacian de&filig;ned by a measure has been shown to be closely related to heat kernel estimates, which under suitable conditions determine whether wave propagates with &filig;nite or in&filig;nite speed. We observe that some self-similar measures de&filig;ned by &filig;nite or in&filig;nite iterated function systems with overlaps satisfy certain “essentially &filig;nite type condition”, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain, under this condition, a closed formula for the spectral dimension of the Laplacian. Earlier results for fractal measures with overlaps rely on Strichartz second-order identities, which are not satis&filig;ed by the measures we consider here. This is a joint work with Wei Tang and Yuanyuan Xie.</p>

Original language	American English
State	Published - Oct 7 2017
Event	Southeastern-Atlantic Regional Conference on Differential Equations (SEARCDE) - Duration: Oct 7 2017 → …

Conference

Conference	Southeastern-Atlantic Regional Conference on Differential Equations (SEARCDE)
Period	10/7/17 → …

Disciplines

Mathematics
Physical Sciences and Mathematics

Keywords

Fractal Laplacians
One-dimensional
Spectral asymptotics

Access to Document

http://conference.kennesaw.edu/searcde/docs/SEARCDE_2017_Program.pdf

Cite this

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title = "Spectral Asymptotics of Some One-Dimensional Fractal Laplacians",

abstract = " The spectral dimension of the Laplacian de&filig;ned by a measure has been shown to be closely related to heat kernel estimates, which under suitable conditions determine whether wave propagates with &filig;nite or in&filig;nite speed. We observe that some self-similar measures de&filig;ned by &filig;nite or in&filig;nite iterated function systems with overlaps satisfy certain “essentially &filig;nite type condition”, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain, under this condition, a closed formula for the spectral dimension of the Laplacian. Earlier results for fractal measures with overlaps rely on Strichartz second-order identities, which are not satis&filig;ed by the measures we consider here. This is a joint work with Wei Tang and Yuanyuan Xie.",

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TY - CONF

T1 - Spectral Asymptotics of Some One-Dimensional Fractal Laplacians

AU - Ngai, Sze-Man

PY - 2017/10/7

Y1 - 2017/10/7

N2 - The spectral dimension of the Laplacian de&filig;ned by a measure has been shown to be closely related to heat kernel estimates, which under suitable conditions determine whether wave propagates with &filig;nite or in&filig;nite speed. We observe that some self-similar measures de&filig;ned by &filig;nite or in&filig;nite iterated function systems with overlaps satisfy certain “essentially &filig;nite type condition”, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain, under this condition, a closed formula for the spectral dimension of the Laplacian. Earlier results for fractal measures with overlaps rely on Strichartz second-order identities, which are not satis&filig;ed by the measures we consider here. This is a joint work with Wei Tang and Yuanyuan Xie.

AB - The spectral dimension of the Laplacian de&filig;ned by a measure has been shown to be closely related to heat kernel estimates, which under suitable conditions determine whether wave propagates with &filig;nite or in&filig;nite speed. We observe that some self-similar measures de&filig;ned by &filig;nite or in&filig;nite iterated function systems with overlaps satisfy certain “essentially &filig;nite type condition”, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain, under this condition, a closed formula for the spectral dimension of the Laplacian. Earlier results for fractal measures with overlaps rely on Strichartz second-order identities, which are not satis&filig;ed by the measures we consider here. This is a joint work with Wei Tang and Yuanyuan Xie.

KW - Fractal Laplacians

KW - One-dimensional

KW - Spectral asymptotics

UR - https://digitalcommons.georgiasouthern.edu/math-sci-facpres/368

UR - http://conference.kennesaw.edu/searcde/docs/SEARCDE_2017_Program.pdf

M3 - Presentation

T2 - Southeastern-Atlantic Regional Conference on Differential Equations (SEARCDE)

Y2 - 7 October 2017

ER -

Spectral Asymptotics of Some One-Dimensional Fractal Laplacians

Abstract

Conference

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