The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type

Convergence theorems and asymptotic estimates (as ε → 0) are proved for the eigenvalues and the eigenfunctions of the Neumann problem in a dense singular junction Ω _ɛ of a domain Ω₀ and a large number N of thin cylinders with thickness of order ε=lN⁻¹, where l is the total length of common boundaries for Ω₀ and the cylinders in question. Bibliography: 27 titles. We dedicate the present paper to Olga Arsenievna Oleinik, as a symbol of our deep respect and gratitude Translated from Trudy Seminara imeni I G. Petrovskogo, No. 19. pp. 000-000. 0000.