= Poisson equation on a disk with angular forcing
{c}
{title2=$R_n(r)=\frac{r^n-r^2}{n^2-4}\quad(n\ne2)$}
For the <Poisson equation> $\Delta u=\sum_{n\geq1}b_n\sin(n\theta)$ on the unit disk with zero <Dirichlet boundary condition>, <separation of variables> in <polar coordinates> gives $r^2R_n''+rR_n'-n^2R_n=r^2$. For $n\ne2$, boundedness at $r=0$ and $R_n(1)=0$ select $R_n=(r^n-r^2)/(n^2-4)$. The resonant mode is $R_2=r^2\log r/4$. For discontinuous angular data, the <Poisson equation> is interpreted in its <weak formulation>; its <Dirichlet energy> gives uniqueness. <Orthogonality> of the angular <Fourier modes> gives $\int u f\,dA=-\sum_n\pi b_n^2/[4(n+2)^2]$, with the resonant case obtained directly or by the limit $n\to2$.
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