TY - JOUR
T1 - Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums
AU - Chen , Hua
AU - Li , Jinning
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 - <p style="text-align: justify;">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.</p>