18.3.1.1 Solution ppc-a

Question:

Show that the Dirichlet boundary-value problem for the Poisson equation on a finite domain,

$$\nabla^2 u = f \quad\mbox{on}\quad\Omega\qquad u = g \quad\mbox{on}\quad\delta\Omega$$

has unique solutions. You cannot have two different solutions $u_1$ and $u_2$ to this problem.

Answer:

Suppose that there are two solutions $u_1$ and $u_2$. For nonuniqueness, the difference $v$ $\vphantom0\raisebox{1.5pt}{$=$}$ $u_2-u_1$ must be nonzero.

Show that $v$ satisfies the Laplace equation by subtracting the Poisson equations satisfied by $u_1$ and $u_2$. Show that $v$ is zero on the boundaries by subtracting the boundary conditions satisfied by $u_1$ and $u_2$.

Now use the maximum and minimum properties of the Laplace equation to show that $v$ is zero. That means that $u_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $u_1+v$ equals $u_1$. So the supposed two different solutions are not different.