Asymptotic of the spectrum of mixed boundary value problem in a thick periodic junction of type 3:2:2 (Q2767621)

Let us consider the domain \(\Omega_{\varepsilon}=\Omega_0\cup G(\varepsilon)\), \(\Omega_0=\{x\in\mathbb{R}^3:0<x_1<a\), \(0< x_2<\gamma(x_1)\), \(0<x_3<d\}\), \(G(\varepsilon)=\bigcup_{j=0}^{N-1}G_{j}(\varepsilon)\), \(G_{j}(\varepsilon)=\{x:\;|x_1-\varepsilon(j+1/2)|<\varepsilon h/2\), \(-f(\varepsilon(j+1/2))<x_2<0\), \(0<x_3<d\}\), where \(\gamma\) is a positive continuous differentiable function on \([0,a]\); \(h\in (0,1)\), \(\varepsilon=a/N\); \(N\) is a sufficiently large positive integer; \(f\in C^1([0,a])\). The domain \(\Omega_{\varepsilon}\) is called junction of type 3:2:2. In \(\Omega_{\varepsilon}\) the following spectral problem is considered: \(-\Delta_{x} u(\varepsilon,x)=\lambda(\varepsilon)u(\varepsilon,x)\), \(x\in \Omega_{\varepsilon}\), \(u(\varepsilon,x)=0\), \(x\in Q_{\varepsilon}\), \(\partial_{\nu}u(\varepsilon,x)=0\), \(x\in \Omega_{\varepsilon}\setminus Q_{\varepsilon}\), where \(\partial_{\nu}=\partial/\partial\nu\) is the external normal derivative, \(Q_{\varepsilon}=\bigcup_{j=0}^{N-1}\{x: |x_1-\varepsilon(j+1/2))|<\varepsilon h/2\), \(x_2=-f(\varepsilon(j+1/2))\), \(0<x_3<d\}\). The asymptotic behaviour, as \(\varepsilon\to 0\), of eigenvalues and eigenfunctions of the considered problem is investigated.