18.4.1.1 Solution emp-a

Question:

Show that the Poisson equation

$\begin{displaymath} \nabla^2 u = f \end{displaymath}$

with boundary conditions

$\begin{displaymath} u_y(x,1)=g_1(x) \qquad u_y(x,0)=g_2(x) \end{displaymath}$

$\begin{displaymath} u(0,y)=g_3(y) \qquad u(1,y)+u_x(1,y)=g_4(y) \end{displaymath}$

has unique solutions.

Answer:

Follow the lines of the uniqueness proofs above. However, in this case you need to write out all four parts of the boundary integral separately. Then you can follow arguments like the ones in the text to show that the diffence between any two solutions is still a constant, and that that constant is still zero.