21. Solve the Neumann problem for a rectangle:
$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$, $0 < x < a$, $0 < y < b$
$\frac{\partial u}{\partial y}|_{y=0} = 0$, $\frac{\partial u}{\partial y}|_{y=b} = 0$, $0 < x < a$
$\frac{\partial u}{\partial x}|_{x=0} = 0$, $\frac{\partial u}{\partial x}|_{x=a} = g(y)$, $0 < y < b$.
(a) Explain why a necessary condition for a solution $u$ to exist is that $g$ satisfy
$\int_0^b g(y)dy = 0$.
This is sometimes called a compatibility condition. Do some extra reading and explain the
compatibility condition on physical grounds.
(b) If $u$ is a solution of the BVP, explain why $u + c$, where $c$ is an arbitrary constant, is also a solution.
