In $\mathbb{R}^{3}$ show that, within a closed surface $S$, there is at most one solution of Poisson's equation, $\nabla^{2} \phi=\rho$, satisfying the boundary condition on $S$

$$\alpha \frac{\partial \phi}{\partial n}+\phi=\gamma \text {, }$$

where $\alpha$ and $\gamma$ are functions of position on $S$, and $\alpha$ is everywhere non-negative.

Show that

$$\phi(x, y)=e^{\pm l x} \sin l y$$

are solutions of Laplace's equation $\nabla^{2} \phi=0$ on $\mathbb{R}^{2}$.

Find a solution $\phi(x, y)$ of Laplace's equation in the region $0<x<\pi, 0<y<\pi$ that satisfies the boundary conditions

$$\begin{array}{cccc} \phi=0 & \text { on } & 0<x<\pi & y=0 \\ \phi=0 & \text { on } & 0<x<\pi & y=\pi \\ \phi+\partial \phi / \partial n=0 & \text { on } & x=0 & 0<y<\pi \\ \phi=\sin (k y) & \text { on } & x=\pi & 0<y<\pi \end{array}$$

where $k$ is a positive integer. Is your solution the only possible solution?