Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums (Analysis in Theory and Applications, 2021)
Let $Ω$, with finite Lebesgue measure $|Ω|$, be a non-empty open subset of $\mathbb{R}$, and $Ω=\bigcup_{j=1}^∞Ω_j$, where the open sets $Ω_j$ are pairwise disjoint and the boundary $Γ=\partialΩ$ has Minkowski dimension $D\in (0,1)$. In this paper we study the Dirichlet eigenvalues problem on the domain $Ω$ 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.