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Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals
Authors:Pablo Amster
Institution:a Universidad de Buenos Aires, Pab. I Ciudad Universitaria (1428), Buenos Aires, Argentina
b Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, Los Polvorines (1613), Prov. Buenos Aires, Argentina
Abstract:Let Ta,b] be a time scale with a,bT. In this paper we study the asymptotic distribution of eigenvalues of the following linear problem −uΔΔ=λuσ, with mixed boundary conditions αu(a)+βuΔ(a)=0=γu(ρ(b))+δuΔ(ρ(b)). It is known that there exists a sequence of simple eigenvalues k{λk}; we consider the spectral counting function View the MathML source, and we seek for its asymptotic expansion as a power of λ. Let d be the Minkowski (or box) dimension of T, which gives the order of growth of the number K(T,ε) of intervals of length ε needed to cover T, namely K(T,ε)≈εd. We prove an upper bound of N(λ) which involves the Minkowski dimension, N(λ,T)?Cλd/2, where C is a positive constant depending only on the Minkowski content of T (roughly speaking, its d-volume, although the Minkowski content is not a measure). We also consider certain limiting cases (d=0, infinite Minkowski content), and we show a family of self similar fractal sets where N(λ,T) admits two-side estimates.
Keywords:Asymptotic of eigenvalues  Lower bounds  Time scales
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