Poisson equation on a square

Question

Studying PDEs from the notes of my professor, and there's a part I don't understand about seeking a solution for the Poisson equation on a square. Let's start from the beginning though. We want to solve the problem $$\begin{cases} &\nabla^2u=F(x,y) \\ &u(x,0)=a(x) \\ &u(\pi,y)=b(y) \\ &u(x,\pi)=c(x) \\ &u(0,y)=d(y) \end{cases}$$ By linearity, we can solve 5 problems distinctly. Among these 5 problems, there's the "seeking a solution for the nonhomogeneous equation". At one point, it says on the notes: We can write $F(x,y)$ as $\sum F_n(y)X_n(x)$, where $X_n(x)$ satisfies the equation $X_n''(x)=-\lambda X_n(x)$. I know he wants to solve it using Lagrange's method, but I don't quite understand why $F(x,y)$ can be written that way. Anyone?

Answer 1

The reason this is possible is that the eigenfunctions (the functions that satisfy $X''=-\lambda X$ and the boundary conditions) form a basis of $L^2[0,\pi]$. Therefore, for each $y$ one can expand $F(x,y)$ in this basis; the coefficients become $F_n(y)$.

These coefficients come from integral formulas like $\int_0^\pi F(x,y)X_n(x)\,dx$, so they inherit continuity, or smoothness, from $F$.

But often, a lecturer doesn't have time or inclination to go into the details of existence (what are the function spaces, why is this a basis, etc.), so the exposition becomes "let's assume this is possible, and see what we get".