# First eigenvalue of the spherical cap

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 $$\lambda(r)$$ ?

For $$n=3$$, $$\lambda(r)=\frac{\pi^2}{r^2}-1,\ 0
Unfortunately, the first Dirichlet eigenvalue for spherical caps in other dimensions cannot be calculated explicitly, but some estimates are known: $$\lambda(r)=\frac{j_{(n-2)/2,1}^2}{r^2}+O(1)\quad\text{as } r\to0^+,$$ where $$j_{\nu,1}$$ is the first zero of Bessel function $$j_\nu$$, and as $$r\to\pi^{-}$$, $$\lambda(r)=\begin{cases}c_{n}\left(\pi-r\right)^{n-2}+o\left(\left(\pi-r\right)^{n-2}\right),\quad & n\geq3;\\ -c_{2}\log^{-1}(\pi-r)+o\left(\log(\pi-r)^{-1}\right),\quad &n=2. \end{cases}$$ for some constants $$c_n$$. The proof and more sharper estimates on $$\lambda(r)$$ can be found in a paper by Borisov and Freitas.