Consider, on a rectangle ($0<x<L,\ 0<y<H$) $$\begin{cases} \triangledown^2u=Q(x,y), \text{on $D$}\\ \triangledown u \cdot\vec{n}=0\ \text{on $\partial D$} \end{cases}$$ Solve using the method of eigenfunction expansion.
So we wish to use the method of eigenfunction expansion. I am however stuck using the boundary conditions for this problem. So the idea is that we use the method of seperation of variables, which results in:
$$\begin{cases} \triangledown^2\phi=-\lambda\phi\ o\text{n}\ D\\ \phi\cdot\vec{n}=0\ \text{on}\ \partial D \end{cases} \\ u(x,y)=\sum_ia_i\phi_i$$
Could anyone give me a hint on how to continue this problem? I presume I need the Divergence theorem, which states (for this problem):
$$-\lambda\iint_D\phi dV=\iint_D \triangledown^2\phi dV\\=\oint_{\partial D}\triangledown\phi\cdot\vec{n}dV=0$$ But I am not sure how to continue with this result.
Begin with $\nabla^2\phi = \lambda \phi$. Use separation of variables in $xy$-plane. Then you get the set of eigenfunctions. Write most general form of expansion and impose boundary conditions. Then construct Green function in terms of eigenfunctions. With the Green function, and $Q(x,y)$ as source, find solution $u(x,y)$.