Asymptotic analysis of spectral problems in thick junctions with the branched fractal structure. (6th September 2022) - Mel'nyk, Taras A.
Abstract : A spectral problem is considered in a domain Ω ε $$ {\Omega}_{\varepsilon } $$ that is the union of a domain Ω 0 $$ {\Omega}_0 $$ and a lot of thin trees situated ε $$ \varepsilon $$ ‐periodically along some manifold on the boundary of Ω 0 . $$ {\Omega}_0. $$ The trees have finite number of branching levels. The perturbed Robin boundary condition ∂ ν u ε + ε α i k i, m u ε = 0 $$ {\partial}_{\nu }{u}^{\varepsilon }+{\varepsilon}^{\alpha_i}{k}_{i, m}{u}^{\varepsilon }=0 $$ is given on the i $$ i $$ th branching layer; { α i } $$ \left\{{\alpha}_i\right\} $$ are real parameters. The asymptotic analysis of this problem is made as ε → 0 $$ \varepsilon \to 0 $$, that is, when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the Hausdorff convergence of the spectrum to the spectrum of the corresponding nonstandard homogenized spectral problem is proved, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.