Let $S$ be the round $n$-sphere of radius $R$ in Euclidean space, and let $r$ be the intrinsic distance from the north pole. Further, let $U(r)$ be the spherical cap of intrinsic radius r. (So $U(0)$ is the north pole and $U(\frac{\pi R}{2})$ is the upper hemisphere.)

Let $\lambda(r)$ be the first eigenvalue of the Laplacian on $U(r)$, with Dirichlet boundary condition. What is the expression of $\lambda(r)$ ?