Characterization of the eigenfunctions in the singularly perturbed domain (Q1112257)

We consider the eigenvalue problem of the Laplacian with Neumann boundary condition, \[ \Delta \Phi +\mu \Phi =0\text{ in } \Omega (\zeta),\quad \partial \Phi /\partial \nu =0\text{ on }\partial \Omega (\zeta). \] We deal with a partial degeneration of a bounded domain of some special type, that is, the domain is expressed as \(\Omega (\zeta)=D_ 1\cup D_ 2\cup Q(\zeta)\) where \(Q(\zeta)\) degenerates and converges to a 1-dimensional line segment in some way as \(\zeta\to 0\). This domain is often called the dumbell shaped domain. Our purpose to give some elaborate characterization of the eigenfunctions, especially we study the behaviors on the degenerating portion \(Q(\zeta)\). We show that the set of the orthonormalized eigenfunctions \(\{\Phi_{k,\zeta}\}^{\infty}_{k=1}\) is divided as follows: \(\{\Phi_{k,\zeta}\}^{\infty}_{k=1}= \{\phi_{k,\zeta}\}^{\infty}_{k=1}\cup \{\psi_{k,\zeta}\}^{\infty}_{k=1},\) where the elements of the right-hand part are associated with the fixed part \(D_ 1\cup D_ 2\) and the degenerating part \(Q(\zeta)\), respectively. For each \(k\geq 1\), \(\|\phi_{k,\zeta}\|_{L^{\infty}(\Omega (\zeta))}\) is bounded in \(\zeta >0\) and \(\|\psi_{k,\zeta}\|_{L^{\infty}(\Omega (\zeta))}\) tends to infinity as \(\zeta\) \(\to 0\). The asymptotic behavior of \(\phi_{k,\zeta}\) in \(Q(\zeta)\) for small \(\zeta >0\) is characterized by a certain ordinary differential equation in the limit line segment \(L\equiv \lim_{\zeta \to 0}Q(\zeta)\). The details including the proof of the above content and some elaborate properties of \(\psi_{k,\zeta}\) will be published by the author [The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary condition, J. Differ. Equations 77, 322--350 (1989; Zbl 0703.35138), erratum J. Differ. Equations 84, No. 1, 204 (1990)].