On eigenfrequencies of bodies with thin branches. II: Asymptotics
In [Math. Notes 54, No. 6, 1192-1199 (1993); translation from Mat. Zametki 54, No. 6, 10-21 (1993; Zbl 0829.35088)] we investigated eigenfrequencies of \(n\)-dimensional bodies \(\Omega_\varepsilon\) consisting of two components: a fixed bounded volume \(\Omega\) and a thin cylindrical branch \(\kappa_\varepsilon\) adjoined to \(\Omega\) and having finite length \(h\); \(\varepsilon\ll 1\) is the ``diameter'' of the section of \(\kappa_\varepsilon\) (the precise definition of \(\Omega_\varepsilon\) will be given below). The mathematical statement of the problem is reduced to the following boundary value problems on eigenvalues: \[ \begin{alignedat}{4} (\Delta+ \lambda_\varepsilon) \psi_\varepsilon & = 0, &\quad x & \in \Omega_\varepsilon; &\quad \partial\psi_\varepsilon/\partial\nu & = 0, &\quad x & \in \partial\Omega_\varepsilon,\tag{1}\\ (\Delta+ \lambda_\varepsilon)\psi_\varepsilon & = 0, &\quad x & \in \Omega_\varepsilon; &\quad \psi_\varepsilon & = 0, &\quad x &\in \partial\Omega_\varepsilon,\tag{2}\end{alignedat} \] where \(\nu\) is an exterior normal to \(\Omega_\varepsilon\). The boundary value problem (2) corresponds to a fixed body, and the problem (1) to a body with free boundary. The eigenvalues \(\lambda_\varepsilon\) are squares of the corresponding eigenfrequencies. In [loc. cit.] it was shown that when reducing the diameter of the section of the branch all bounded eigenvalues of the problem (2) converge to the set \(\widetilde\Sigma\) of eigenvalues of the Dirichlet problem for the Laplace operator in \(\Omega\). Conversely, if \(\lambda_0\) is a \(q\)-multiple eigenvalue of the Dirichlet problem in \(\Omega\), then the total algebraic multiplicity of eigenvalues of (2) that are converging to \(\lambda_0\) is also equal to \(q\). But for the problem (1) the situation is quite different: as \(\varepsilon\to 0\) the eigenvalues of this problem converge both to the set \(\Sigma^\nu\) of eigenvalues of the Neumann problem for the Laplace operator in the domain \(\Omega\) and to the set \(\Sigma^{ch}= \{\mu_m= (\pi(2m- 1)/(2h))^2\}^\infty_{m= 1}\) which is the set of eigenvalues of the following Sturm-Liouville problem: \[ u''+ \mu u= 0,\quad -h< t< 0,\quad u'(- h)= u(0)= 0. \] In the present paper, using the method of agreement of asymptotic expansions, we construct asymptotics of eigenvalues \(\lambda_\varepsilon\) of the boundary value problems (1) and (2) in \(\mathbb{R}^3\) with respect to the small parameter \(\varepsilon\). We consider the cases, where the limit values of the eigenfunctions are such that \(\lambda_0\in \Sigma^\nu_1\backslash \Sigma^{ch}\), \(\lambda_0 \in \Sigma^{ch}\backslash \Sigma^\nu\) and \(\lambda_0\in \widetilde\Sigma_1\), where \(\Sigma^\nu_1\) and \(\widetilde\Sigma_1\) are the sets of first-order eigenvalues of the Neumann and Dirichlet problems in \(\Omega\), respectively.
Dirichlet problem for the Laplace operator
Neumann problem for the Laplace operator
scientific article; zbMATH DE number 796760