I have to solve Poisson's equation
$\nabla^2u=f(x,y)$
in a domain $\Omega$ whith inhomogeneus boundary conditions.
Some books propose to separate this problem in subproblems and for the inhomogeneus part( $f(x,y)$) to solve the eigenvalue problem \begin{equation} \nabla^2 u=-\lambda u \end{equation} with homogeneus boundary conditions. I cant understand what lead us to solve this eigenvalue problem. Which is the reason that this eigenvalue problem works?
Every answer will be helpful. Thank you very much
The motivation for this method is based on the fact that the eigenfunctions will form a complete and orthogonal set which can be used (with appropriate weights) to represent any "smooth" function.
Assuming you've found you eigenpair $\{\lambda_n,\phi_n\}$. You would see that plugging in $u = \sum_n a_n \phi_n$ leads to $$ \sum_n a_n \nabla^2 \phi_n = -\sum_n a_n \lambda_n \phi_n = f. $$ Using the orthogonality of the eigenfunctions, you can find $a_n$.