Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions. (Q2925951)

The aim of this paper are asymptotic expansions for the eigenvalues \(\lambda_n\) of the Laplacian on a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension. More precisely, let \(\Sigma\) be a connected orientable \(C^2\) hypersurface in \( \mathbb{R}^d, d \geq 2, \) and \(\epsilon > 0 \) a given parameter; then, the eigenvalues of the Laplacian with Dirichlet and Neumann boundary conditions on \(\Sigma\) and \(\Sigma_\epsilon = \Sigma + \epsilon \;n(\Sigma)\), respectively, on the domain NEWLINE\[NEWLINE\Omega_\epsilon = \{ x + \epsilon t n(x) \in \mathbb{R}^d; (x,t) \in \; \Sigma \times (0,1) \},NEWLINE\]NEWLINE are the topic, where \(n\) denotes a unit normal vector field. Let NEWLINE\[NEWLINEH_\epsilon = -\Delta_g + \frac{\kappa}{\epsilon}\quad \text{on } L^2(\Sigma),NEWLINE\]NEWLINE where \(-\Delta_g \) denotes the Laplace-Beltrami operator on \(\Sigma\) with Dirichlet boundary conditions if \(\partial \Sigma\) is not empty and \(\kappa = \kappa_1 +\dots + \kappa_{d-1}\) is a \(d-1\) multiple of the mean curvature on \(\Sigma\) and \(\mu_1(\epsilon), \mu_2(\epsilon),\dots\) the eigenvalues.NEWLINENEWLINENEWLINEThen, the result of the paper is the following theorem.NEWLINENEWLINENEWLINETheorem 1.1 For all \(n\geq 1\), NEWLINE\[NEWLINE\lambda_n(\epsilon) =\left(\frac{\pi}{2\epsilon}\right)^2 + \mu_n(\epsilon) + O(1) \text{ as } \epsilon \to 0.NEWLINE\]NEWLINE The case \(d=2\) was proved by the author in \textit{D. Krejčiřík} [ESAIM, Control Optim. Calc. Var. 15, No. 3, 555--568 (2009; Zbl 1173.35618)]. Theorem 1.1 follows from asymptotic upper and lower bounds for \(\lambda_n(\epsilon)\) with the same leading terms.