Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums

Let ?, with finite Lebesgue measure |?|, be a non-empty open subset of R, and ? =∪~∞?, where the open sets ?are pairwise disjoint and the boundary Γ = ?? has Minkowski dimension D ∈(0, 1). In this paper we study the Dirichlet eigenvalues problem on the domain ? 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.