Localization effect for Dirichlet eigenfunctions in thin non-smooth domains (Q2790638)

For thin plates \(\Omega^\varepsilon =\{(y,z) \mid y\in\omega, -\varepsilon H_-(y)<z<\varepsilon H_+(y)\}\), where \(\omega\subset \mathbb{R}^2\) is a bounded Lipschitz domain, the authors study asymptotic properties as \(\varepsilon\to 0^+\) of the spectral Laplace--Dirichlet problem. The thickness \(H=H_++H_->0\) possesses a strict maximum \(h=H(0)\) at the origin \(\in\omega\). Moreover, \(H\) is assumed to decay quadratically with distance from the origin. For every \(\varepsilon>0\), \((\lambda_k^\varepsilon)_k\) denotes the increasing sequence of all eigenvalues of \(\Delta^\varepsilon\) with multiple eigenvalues repeated in the standard way. The main results assert that, for fixed \(k\), an asymptotic formula \(\lambda_k^\varepsilon = (\pi/h\varepsilon)^2+\mu_k/\varepsilon+O(\varepsilon^{-1/2})\) holds, and that the eigenfunctions \(u_k^\varepsilon\) decay exponentially with distance from the axis \(y=0\). The numbers \(\mu_k\) are the eigenvalues of a limit spectral problem in \(\mathbb{R}^2\); the associated orthonormal eigenfunctions are denoted \(w_k\). Passage from the original problem to the limit problem is done by weighted averaging in the \(z\)-direction and stretching of the variable \(y\) to \(\eta=\varepsilon^{-1/2}y\). Conversely, an ansatz NEWLINE\[NEWLINE w_k(\varepsilon^{-1/2} y)\sin(\pi(z+\varepsilon H_-(y))/\varepsilon H(y)), NEWLINE\]NEWLINE appropriately cut off and normalized, associates with each \((\mu_k,w_k)\) the eigenvalue-eigenfunction family \((\lambda_k^\varepsilon,u_k^\varepsilon)_{0<\varepsilon<\varepsilon(k)}\). Notably, only the first eigenvalue in the transverse \(z\)-direction matters. This is because higher eigenvalues of \(-\partial_z^2\) relate to \(\lambda_{\ell}^\varepsilon\) for some \(\ell >k\), and \(\varepsilon\) falls out of the asymptotic range when \(k\) is replaced by \(\ell\).NEWLINENEWLINEThe final section discusses generalizations to mixed boundary problems and to the existence of spectral gaps in thin infinite layers.