18.3.3.6 Solution ppe-f

Question:

Show that if the Poisson equation

$$\nabla^2 u = f \qquad\mbox{inside}\qquad\Omega$$

with the Neumann boundary condition

$$\frac{\partial u}{\partial n} = g \qquad\mbox{on}\qquad\delta\Omega$$

has a solution, it is not unique.

Answer:

Suppose you have a solution $u_1$. Then $u_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $u_1+C$, where $C$ is any arbitrary constant, is also a solution. The constant differentiates away in both the partial differential equation and boundary condition.