# list of $L^2$ eigenfunctions and eigenvalues on a cylinder with Dirichlet boundary condition.

Using separation of variables, write down a complete list of $$L^{2}$$ eigenfunctions and of eigenvalues for the Laplacian on the cylinder $$D\times[-1,1]$$ with Dirichlet boundary conditions, where D is the 2-dim disk centered at the origin of radius 2.

Also, use this to solve the heat equation $$\frac{\partial u}{\partial t}=\Delta u$$ on this cylinder with homogeneous Dirichlet boundary condition, with initial data $$u(x,y,z,0)=z$$ where $$z$$ is the third coordinate, corresponding to the height of the cylinder.

Any help is appreciated.

Hint: In cylindrical coordinates \begin{align} \Delta_\text{cylind} u = \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r} \right)+\frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} + \frac{\partial^2 u}{\partial z^2}. \end{align} Since you have Dirichlet boundary condition, then we see that \begin{align} \begin{cases} u(2, \theta, z) = 0 & \text{for } -1\leq z \le 1 \text{ and } 0\leq \theta <2\pi,\\ u(r, \theta, \pm 1) = 0 & \text{for } 0\leq r \le 2 \text{ and } 0\leq \theta <2\pi \end{cases}. \end{align}