Bounds for the first eigenvalue of a spherical cap (Q790364)

The purpose of this paper is to study upper and lower bounds, as well as asymptotic behaviour of the lowest eigenvalue \(\lambda\) of the Laplacian of a spherical cap in m-dimensional space, \(m\geq 3\). By using a probabilistic approach it is shown that \[ \lambda \geq 1/\int^{\theta_ 0}_{0}(1/(\sin x)^{m-2})\int^{x}_{0}(\sin t)^{m-2}dt. \] By using a result of Hobson concerning zeros of Legendre functions as well as bounds for \(\lambda\) given previously by Friedland and Hayman, we give asymptotic expansions for \(\lambda\) as the cap tends to the whole sphere and as the radius of the cap tends to zero. These are used to examine the sharpness of our results.