Characteristic frequencies of bodies with thin spikes. I: Convergence and estimates (Q1896976)

The set \(\Omega_\varepsilon\) is an \(n\)-dimensional body \(\Omega\) with thin spike \(\kappa_\varepsilon\) having ``diameter'' \(\varepsilon\) and finite length \(h\). In this paper we consider the following two eigenvalue (EV) problems: \[ (\Delta + \lambda_\varepsilon) \Psi_\varepsilon = 0, \quad x \in \Omega_\varepsilon, \quad \partial \Psi_\varepsilon/ \partial \nu = 0, \quad x \in \partial \Omega_\varepsilon, \tag{1} \] where \(\nu\) is the outer normal to \(\Omega_\varepsilon\) and \[ (\Delta + \lambda_\varepsilon) \Psi_\varepsilon = 0, \quad x \in \Omega_\varepsilon, \quad \Psi_\varepsilon = 0, \quad x \in \partial \Omega_\varepsilon. \tag{2} \] Nontrivial solutions of (2) describe the characteristic oscillations of the fixed body \(\Omega_\varepsilon\) and the solutions of (1) correspond to the characteristic oscillations of the body with free boundary. The square roots \(k_\varepsilon\) of the EV \(\lambda_\varepsilon\) are the characteristic frequencies of the bodies under the types of boundary conditions indicated. We study the question of the convergence of the EV of the boundary problems (1) and (2) as \(\varepsilon \to 0\). It would seem that as \(\varepsilon \to 0\) the EV of (1) and (2) would converge respectively to the sets \(\Sigma^\nu\) and \(\widetilde \Sigma\) of the EV of the Neumann and Dirichlet problems in \(\Omega\). Indeed such convergence holds for (2). However in the case of boundary conditions (1) it turns out that the EV converge either to \(\Sigma^\nu\) or to the set \(\Sigma^{\text{ch}} = \{\mu_m = (\pi (2m - 1)/(2h))^2\}_{m = 1}^\infty\). Firstly, we establish a correspondence between the multiplicities of the EV of the perturbed and limit problems without any restrictions on \(\Sigma^\nu\) and \(\Sigma^{\text{ch}}\) for (1), and secondly, we get estimates for solutions of the boundary problems \[ (\Delta + \lambda) u_\varepsilon = f, \quad x \in \Omega_\varepsilon, \quad \partial u_\varepsilon/ \partial \nu = 0, \quad x \in \partial \Omega_\varepsilon, \] \[ (\Delta + \lambda) u_\varepsilon = f, \quad x \in \Omega_\varepsilon, \quad u_\varepsilon = 0, \quad x \in \partial \Omega_\varepsilon \] for \(\lambda\) close to the corresponding EV \(\lambda_\varepsilon\) which are uniform in \(\varepsilon\), \((\lambda - \lambda_\varepsilon)^{-1}\) in \(L_2 (\Omega_\varepsilon)\).