Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals
Let T ⊂ [a, b] be a time scale with a, b ∈ T. In this paper we study the asymptotic distribution of eigenvalues of the following linear problem - uΔ Δ = λ uσ, with mixed boundary conditions α u (a) + β uΔ (a) = 0 = γ u (ρ (b)) + δ uΔ (ρ (b)). It is known that there exists a sequence of simple eigenv...
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| Autores principales: | , , |
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_0022247X_v343_n1_p573_Amster |
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| Sumario: | Let T ⊂ [a, b] be a time scale with a, b ∈ T. In this paper we study the asymptotic distribution of eigenvalues of the following linear problem - uΔ Δ = λ uσ, with mixed boundary conditions α u (a) + β uΔ (a) = 0 = γ u (ρ (b)) + δ uΔ (ρ (b)). It is known that there exists a sequence of simple eigenvalues {λk}k; we consider the spectral counting function N (λ, T) = # {k : λk ≤ λ}, 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) ≤ C λd / 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. © 2008 Elsevier Inc. All rights reserved. |
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