Volume 36, Issue 3
Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums

Hua Chen & Jinning Li

Anal. Theory Appl., 36 (2020), pp. 243-261.

Published online: 2020-09

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  • Abstract

Let $\Omega$, with finite Lebesgue measure $|\Omega|$, be a non-empty open subset of $\mathbb{R}$, and $\Omega=\bigcup_{j=1}^\infty\Omega_j$, where the open sets $\Omega_j$ are pairwise disjoint and the boundary $\Gamma=\partial\Omega$ has Minkowski dimension $D\in (0,1)$.  In this paper we study the Dirichlet eigenvalues problem on the domain $\Omega$ and give the exact second asymptotic term for the eigenvalues, which is related to the Minkowski dimension $D$. Meanwhile, we give sharp lower bound estimates for Dirichlet eigenvalues for such one-dimensional fractal domains.

  • Keywords

One-dimensional fractal drum, Dirichlet eigenvalues, Pόlya conjecture, Minkowski dimension.

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-36-243, author = {Hua Chen , and Jinning Li , }, title = {Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {3}, pages = {243--261}, abstract = {

Let $\Omega$, with finite Lebesgue measure $|\Omega|$, be a non-empty open subset of $\mathbb{R}$, and $\Omega=\bigcup_{j=1}^\infty\Omega_j$, where the open sets $\Omega_j$ are pairwise disjoint and the boundary $\Gamma=\partial\Omega$ has Minkowski dimension $D\in (0,1)$.  In this paper we study the Dirichlet eigenvalues problem on the domain $\Omega$ and give the exact second asymptotic term for the eigenvalues, which is related to the Minkowski dimension $D$. Meanwhile, we give sharp lower bound estimates for Dirichlet eigenvalues for such one-dimensional fractal domains.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-SU7}, url = {http://global-sci.org/intro/article_detail/ata/18285.html} }
TY - JOUR T1 - Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums AU - Hua Chen , AU - Jinning Li , JO - Analysis in Theory and Applications VL - 3 SP - 243 EP - 261 PY - 2020 DA - 2020/09 SN - 36 DO - http://doi.org/10.4208/ata.OA-SU7 UR - https://global-sci.org/intro/article_detail/ata/18285.html KW - One-dimensional fractal drum, Dirichlet eigenvalues, Pόlya conjecture, Minkowski dimension. AB -

Let $\Omega$, with finite Lebesgue measure $|\Omega|$, be a non-empty open subset of $\mathbb{R}$, and $\Omega=\bigcup_{j=1}^\infty\Omega_j$, where the open sets $\Omega_j$ are pairwise disjoint and the boundary $\Gamma=\partial\Omega$ has Minkowski dimension $D\in (0,1)$.  In this paper we study the Dirichlet eigenvalues problem on the domain $\Omega$ and give the exact second asymptotic term for the eigenvalues, which is related to the Minkowski dimension $D$. Meanwhile, we give sharp lower bound estimates for Dirichlet eigenvalues for such one-dimensional fractal domains.

Hua Chen & Jinning Li. (2020). Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums. Analysis in Theory and Applications. 36 (3). 243-261. doi:10.4208/ata.OA-SU7
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