The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary condition (Q915998)

Let \(\Omega\) (\(\zeta\)) be the singularly perturbed domain of the form \(\Omega (\zeta)=D_ 1\cup D_ 2\cup Q(\zeta)\), where \(D_ 1\), \(D_ 2\) are bounded domains in \({\mathbb{R}}^ n\), \(\bar D_ 1\cap \bar D_ 2=\emptyset\) and Q(\(\zeta\)) \((\zeta >0)\) is a moving domain which partially degenerates into a line segment \(L\subset R\) as \(\zeta \to +0\). The author gives a characterization of the asymptotic behavior of the eigenfunctions of the Laplace operator in \(\Omega\) (\(\zeta\)) for the Neumann boundary condition on \(\partial \Omega (\zeta)\) as \(\zeta \to +0\). The rate of the degeneration of the function space \(L^ 2(\Omega)\zeta))\), which is associated with the partial degeneration of the domain \(\Omega\) (\(\zeta\)) is also given. The results obtained are applicable to the solutions of some reaction-diffusion equations and systems.