Singular perturbations of curved boundaries in three dimensions. The spectrum of the Neumann Laplacian (Q537679)

Let \(\Omega\) be a domain in \(\mathbb R^3\), \(\overline{\omega}_h\) -- a small cavern with a diameter \(O(h)\) in a boundary \(\partial\Omega\) or inside of the domain \(\Omega\) at the distance \(O(h)\) from \(\partial\Omega\), \(\Omega(h)\) -- a domain \(\Omega\) with a hole \(\overline{\omega}_h\), \(\partial\Omega(0)=\partial\Omega\). There is considered the spectral Neumann problem in perturbed domain \[ -\Delta_xu^h(x)=\lambda^h u^h(x), \quad x \in \Omega(h), \;\partial_{n^h}u^h(x)=0, \;x\in \partial\Omega(h), \tag{1} \] where \(\Delta_x\) is the Laplace operator, \(\partial_{n^h}\) -- the derivative with respect to normal to \(\partial\Omega(h)\). The authors prove the asymptotic expansion formulas with respect to \(h\) of the eigenvalues \(0=\lambda^h_0<\lambda^h_1\leq \lambda^h_2 \leq \cdots \leq \lambda^h_m \cdots \), \(\lambda^h_m \to \infty\) as \(m \to \infty\) and eigenfunctions \(u^h_0, u^h_1, u^h_2, \dots, u^h_m, \dots \) to problem (1), the principle terms of them are the eigenvalues \( 0=\lambda^0_0<\lambda^0_1\leq \lambda^0_2 \leq \cdots \leq \lambda^0_m \cdots \), \(\lambda^0_m \to \infty\) as \(m \to \infty\) and eigenfunctions \(v^0_0, v^0_1, v^0_2, \dots, v^0_m, \dots \) respectively of the unperturbed Neumann problem \[ -\Delta_xv^0(x)=\lambda^0 v^0(x), \quad x \in \Omega, \;\partial_{n}v^0(x)=0, \;x\in \partial\Omega, \] where \(\partial_n\) is the normal derivative to \(\partial\Omega\). In particular, it follows from these formulas that \(\lambda^h_m = \lambda^0_m + O(h^3)\), \(u^h_m=v^0_m+O(h^3)\) as \(h\to 0\).