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, P¤îlya 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.