SFB 1313 Publication "Spectral problems with perturbed Steklov conditions in thick junctions with branched structure"

New SFB 1313 publication, published in "Applicable Analysis". The work has been developed by researchers involved in Project I-08 and external collaborators.

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

Authors

• Taras Mel’nyk (University of Stuttgart, research project I-08)  
•  Tiziana Durante (Università degli Studi di Salerno)

Abstract< 1 , which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0 , i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε .

The article examines spectral problems in a domain Ω_ε, which is the union of a domain Ω₀ and a lot of thin trees that are ε-periodically situated along a manifold on the boundary of Ω₀. The trees possess a finite number of branching levels. At the boundaries of branches from each branching level, the perturbed Steklov spectral condition ∂_νu^ε=ε^αλ^εϱ_i,mu^ε is given, where α ϵ 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.