4)I think you made a mistake. That is not the solution to:
5)The boring what is to substitute that into the PDE and verify that it solves it, i.e. adds up to zero. A more exciting way is to use the Cauchy-Riemann equations. The prinipal logarithm is holomorphic on . That means are harmonic functions.
We can write,
That means the real part is solves Laplace's equation.
3)I am really not sure about this one, but I think I know what I am doing. First, in your problem you have the Dirichlet problem in 3 variables, is that a mistake? I will do it in two variables.
Seperate variables and write .
Then by substituting this into the PDE we have:
Divide through by to get:
And hence by the seperation of variables techinque we have a system of ODE:
Look at the first equation together with the boundary value problem implies that for means non-trivial solutions can only exist iff , that is the solutions are sines and cosines.
Look at the second equation by similar reasoning . So if (by the first equation) it must mean that for otherwise and that will lead to trivial solutions to the two point boundary value problem.