Let $D$ be the unit disk in $\mathbb{R}^2$ and consider the nonhomogeneous Poisson problem
$$ \begin{cases} ∆u = 4 & \text{ in }D \\ u = 0 & \text{ on }∂Ω \\ \end{cases} $$ where we assume that $|u(0, θ)| < +∞$.
How do I find a solution $u(r, θ)$ in terms of the Fourier-Bessel Series of the constant function $f(r, θ) ≡ 4$ ?
Then since $f(r, θ)$ is radial, i.e. independent of $θ$, we know that $u(r, θ)$ must also be radial. How can I use this to find an explicit formula for $u(r, θ) = u(r)$?
To solve the Poisson equation $\nabla u = p$ with some polynomial $p$, it is natural to look for $u$ as a polynomial of degree $\deg p +2$, with undetermined coefficients. Here this means $u$ should be quadratic. There are many polynomial that work, specifically $$u(x,y) = Ax^2+Bxy+Cy^2+Dx+Ey+F,\quad A+C=2$$ Considering the boundary condition, we need $u$ to be radially symmetric. This calls for $B=E=E=0$ and $A=C$. Hence, $u(x,y)=x^2+y^2+F$, and the boundary condition forces $F=-1$.