Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums

LetΩ,with finite Lebesgue measure|Ω|,be a non-empty open subset of R,andΩ=∪∞j-1Ωj,where the open setsΩj are pairwise disjoint and the boundaryГ=■Ωhas Minkowski dimension D∈(0,1).In this paper we study the Dirichlet eigenvalues problem on the domainΩand 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.