Asymptotic Analysis of Spectral Problems in Thick Multi-Level Junctions
Spectral boundary-value problems are considered in a new kind of perturbed domain, namely, thick multi-level junctions. Boundary-value problems in thick one-level junctions (thick junctions) have been intensively investigated recently (see, for instance, [BlGaGr07], [BlGaMe08], [Me08] and, the references there). In [MeNa97]–[Me(3)01], classification of thick one-level junctions was given and basic results were obtained both for boundary-value and spectral problems in thick junctions of different types. It was shown that qualitative properties of solutions essentially depend on the junction type and on the conditions given on the boundaries of the attached thin domains. It is known that the asymptotic behavior of the spectrum of a perturbed spectral problem is highly sensitive to perturbation, and it is unexpected. This was also observed for spectral problems in thick junctions with Neumann conditions ([MeNa97] and [Me00]), with Dirichlet conditions ([Me99] and [Me(3)01]), with Fourier conditions ([Me(2)01]) and with Steklov ones ([Me(1)01]).
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- [Me(2)01]Mel'nyk, T.A.: Asymptotic behaviour of eigenvalues and eigenfunctions of the Fourier problem in a thick junction of type 3:2:1, in Grouped and Analytical Methods in Mathematical Physics, Academy of Sciences of Ukraine, Kiev (2001), 187-196.Google Scholar