7.9.2. (c) Solve Laplace's equation inside a semicircular cylinder, subject to the boundary conditions
$$ \dfrac{\partial}{\partial z} u (r, \theta, 0) = 0, \quad \dfrac{\partial}{\partial z} u (r, \theta, H) = 0, \quad \dfrac{\partial}{\partial \theta} u (r, 0, z) = 0, \quad \dfrac{\partial}{\partial \theta} u (r, \pi, z) = 0, \quad \dfrac{\partial}{\partial r} u (a, \theta, z) = \beta (\theta, z) $$ Under what condition does a solution exist?
How can I know a solution exists? My intuition tells me flux in should equal flux out, but I don't know how can I get a mathematical condition for that.