As I am not an expert on biharmonic and more in general polyharmonic equation. Is there a procedure similar to the one highlighted here to derive the green function?
Some people however don't seem to follow that procedure to define what a Green function is. Many seem to simply define it as the solution of
$$ LG(x,y) = \delta(x - y) $$
where $L$ is any linear differential operator. I can have an intuition why defining the Green function as above make sense. Suppose indeed we want to solve a problem of the form
$$ Lu = f $$
And suppose $G$ is the Green function, defined earlier. Assuming we're working with distributions We then have $$ Lu = f = f * \delta = f * LG = L(f*G) $$ which yields $$ Lu = L(f*G) \Rightarrow Lu - L(f*G) = L(u - f*G) = 0 $$ therefore if $u - f*G = \Phi$ (the fundamental solution for the operator $L$ then we have $$ u = \Phi + f*G. $$ Which would be the solution. In the link I gave the derivation is taken from Evans - Partial Differential Equations and it seems a bit more involved since it also takes into account boundary conditions to be applied.
Assuming the math I did in this post is correct is there a reason why Evans prefers to follow a different method for his derivation?
If the math I did is wrong what is the right way to derive the Green function for at least $L = \Delta^2$ or more in general $L = \Delta^k$?