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Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions. (English) Zbl 1340.35239
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 \[ \Omega_\epsilon = \{ x + \epsilon t n(x) \in \mathbb{R}^d; (x,t) \in \; \Sigma \times (0,1) \}, \] are the topic, where \(n\) denotes a unit normal vector field. Let \[ H_\epsilon = -\Delta_g + \frac{\kappa}{\epsilon}\quad \text{on } L^2(\Sigma), \] 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.
Then, the result of the paper is the following theorem.
Theorem 1.1 For all \(n\geq 1\), \[ \lambda_n(\epsilon) =\left(\frac{\pi}{2\epsilon}\right)^2 + \mu_n(\epsilon) + O(1) \text{ as } \epsilon \to 0. \] The case \(d=2\) was proved by the author in 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.

MSC:
35P15 Estimates of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
81Q15 Perturbation theories for operators and differential equations in quantum theory
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