The asymptotics of the solution to the Neumann spectral problem in a domain of the ``dense-comb'' type
The authors study a Neumann spectral problem for the Laplacian in a domain which consists of a fixed open set \(\Omega\) of \({\mathbb R}^2\) (the ``body'') to which are glued a large number \(N\) of thin rectangles (the ``teeth'') of length 1 and of width \(\varepsilon \sim N^{-1}\). Then they construct asymptotic solutions to this problem as \(\varepsilon\rightarrow 0\), using a particular matching method at the junction between \(\Omega\) and the teeth which lead to a spectral problem with discontinuous coefficients. Finally, they show that these constructions coincide to the actual solutions up to an error of order \({\mathcal O}(\varepsilon^{{1\over 2} -\delta})\) for any positive \(\delta\).
scientific article; zbMATH DE number 1259428