Kind of a long shot, but is there a reference out there regarding the Green's function for $$\frac{d^4}{dx^4}-2\frac{d^2}{dx^2}+Id=(\frac{d^2}{dx^2}-Id)^2$$ for boundary conditions $f(0)=f(1)=f''(0)=f''(1)=0$?
The reason I'm looking for this is that I'm trying to see if I can find bounds for $f'(0)$ and $f'(1)$ (i.e. at the end points) in terms of the inhomogeneous term by differentiating and estimating the Green's function.