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Abstract:

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.

Registro:

Documento: Artículo
Título:Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals
Autor:Amster, P.; De Nápoli, P.; Pinasco, J.P.
Filiación:Universidad de Buenos Aires, Pab. I Ciudad Universitaria, 1428 Buenos Aires, Argentina
Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, Los Polvorines (1613), Prov. Buenos Aires, Argentina
Palabras clave:Asymptotic of eigenvalues; Lower bounds; Time scales
Año:2008
Volumen:343
Número:1
Página de inicio:573
Página de fin:584
DOI: http://dx.doi.org/10.1016/j.jmaa.2008.01.070
Título revista:Journal of Mathematical Analysis and Applications
Título revista abreviado:J. Math. Anal. Appl.
ISSN:0022247X
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v343_n1_p573_Amster

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Citas:

---------- APA ----------
Amster, P., De Nápoli, P. & Pinasco, J.P. (2008) . Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals. Journal of Mathematical Analysis and Applications, 343(1), 573-584.
http://dx.doi.org/10.1016/j.jmaa.2008.01.070
---------- CHICAGO ----------
Amster, P., De Nápoli, P., Pinasco, J.P. "Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals" . Journal of Mathematical Analysis and Applications 343, no. 1 (2008) : 573-584.
http://dx.doi.org/10.1016/j.jmaa.2008.01.070
---------- MLA ----------
Amster, P., De Nápoli, P., Pinasco, J.P. "Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals" . Journal of Mathematical Analysis and Applications, vol. 343, no. 1, 2008, pp. 573-584.
http://dx.doi.org/10.1016/j.jmaa.2008.01.070
---------- VANCOUVER ----------
Amster, P., De Nápoli, P., Pinasco, J.P. Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals. J. Math. Anal. Appl. 2008;343(1):573-584.
http://dx.doi.org/10.1016/j.jmaa.2008.01.070