Spectral problems with perturbed Steklov conditions in thick junctions with branched structure

Abstract

The article examines spectral problems in a domain Ω ε , which is the union of a domain Ω 0 and a lot of thin trees that are ε-periodically situated along a manifold on the boundary of Ω 0 . The trees possess a finite number of branching levels. At the boundaries of branches from each branching level, the perturbed Steklov spectral condition ∂ ν u ε = ε α λ ε ϱ i , m u ε is given, where α ∈ R . For a fixed ε, the problem possesses a discrete spectrum, and we analyze the asymptotic behavior of the eigenvalues and eigenfunctions as ε → 0 , i.e. when the number of thin trees infinitely increases and their thickness vanishes. Three qualitatively distinct cases of the asymptotic behavior of the spectrum are identified depending on the parameter α. For each case, the study provides proof of the Hausdorff convergence of the spectrum, construction of leading asymptotic terms, and justification of the asymptotic estimates for the eigenvalues and eigenfunctions. For a point in the essential spectrum of the homogenized problem, we discover finite-energy eigenoscillations localized in one or some of the thin trees, and prove the asymptotic estimates. Cases, where the Steklov condition contains α depending on the branching level, are also discussed.