I came across the following boundary value problem that I can't solve. It's the Laplacian on the upper half of an annulus with radius $1 \leq r \leq 2$ in polar coordinates:
$u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta \theta} =0$
$u(1,\theta)=u(2,\theta)=0, ~~~ 0<\theta<\pi$
$u(r,0)=0, ~~ u(r,\pi)=r, ~~~ 1
The problem mentions that one must show in this case the choice of separation constant $\lambda=-\alpha^2<0$ leads to eigenvalues and eigenfunctions. The problem is that in this case, after separating variables and solving
$ \Theta'' + \lambda \Theta = 0$
$ r^2R'' + rR' - \lambda R =0$
we would get $\Theta(\theta)=c_1 \cosh(\alpha \theta) + c_2 \sinh(\alpha \theta)$ which is not periodic in $\theta$. Usually we always have $\alpha^2>0$ which gives us the periodic sine and cosine solution for $\Theta$. Help would be appreciated!