Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums

Let Ω,with finite Lebesgue measure |Ω|,be a non-empty open subset of R,and Ω = ∪∞j=1 Ωj,where the open sets Ωj are pairwise disjoint and the boundaryΓ = ?Ω has Minkowski dimension D ∈(0,1).In this paper we study the Dirich-let 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.