Eigenvalues of the Laplace-Beltrami operator on a large spherical cap under the Robin problem
This paper studies the linear eigenvalue problem \[ \Lambda v+\lambda v=0\quad \text{in } \Omega_\epsilon \subset S^n, \] \[ (\cos \sigma )\partial_n v+(\sin \sigma )v=0\quad \text{on } \partial \Omega_\epsilon, \] where \(\Lambda \) is the Laplace-Beltrami operator on the unit sphere \(S^n \subset R^{n+1}\) for \(n\geq 2\), \(\Omega_\epsilon \) is the spherical cap centered at the north pole \((0,0,\dots,0,1)\) of its geodesic radius \(\pi -\epsilon \) with \(\epsilon >0\), \(\partial_n\) is the derivative of the outer unit normal to \(\Omega_\epsilon \) and \(\sigma \in [0,\pi /2]\), \(\lambda >0\) are parameters. The behaviour of the eigenvalues of \(-\Lambda \) as \(\epsilon \to 0_+\) is studied.
scientific article; zbMATH DE number 6387908