18.4.1.2 Solution emp-b

Question:

Using the arguments given in the text, uniqueness can not be shown for the Poisson equation

$$\nabla^2 u = f$$

with boundary conditions

$$u_y(x,1)=g_1(x) \qquad u_y(x,0)=g_2(x)$$

$$u(0,y)=g_3(y) \qquad u(1,y)-u_x(1,y)=g_4(y)$$

Of course, just because you cannot prove uniqueness does not mean it is not true. But show that this problem never has unique solutions. If it has a solution at all, there are infinitely many different ones.

Answer:

If you can show that the homogeneous problem has a nontrivial (nonzero) solution $v$, you are done. Then if there is any solution $u_1$ to the original problem, infinitely many more solutions can be obtained by adding arbitrary multiples of $v$ to $u_1$.

To find a nontrivial solution, guess it. In particular, based on the boundary conditions for $v$ at $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 and $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, guess that the nontrivial solution $v$ may be independent of $x$. If you plug that assumption into the partial differential equation and boundary conditions, you can indeed find a nonzero solution.

If you solve the problem for a general mixed boundary condition, using separation of variables, you find that for many values of the coefficients $A$ and $B$, but not all, there are nonunique solutions. However, there are none unless $A$ and $B$ have opposite sign.