For the 2-sphere $\mathbb{S}^2$, the first eigenvalue of the spherical cap can be calculated via stereographic projection. Under this projection, $U(r)$ is a ball $B_{\mathbb{C}}(0,\tan(r/2))$ in $\mathbb{C}$, whose first eigenvalue is $$\lambda_{1}(r)=\left(\frac{\mu_1}{\tan(r/2)}\right)^{2},$$ where $\mu_1$ is the fist zero of the Bessel function $$J_{0}(t)=\frac{1}{\pi}\int_{0}^{\pi}\cos(t\sin(\theta))\ \mathrm{d}\theta$$ and $\mu_1\approx2.4048$.

For $n=3$, $$\lambda(r)=\frac{\pi^2}{r^2}-1,\ 0<r\leq\pi.$$

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.