I have to find the exact solution of the following: $$-\nabla^{2}{u}=1$$ on the two dimensional domain $$\Omega=\{z=(x,y):|z|\ge{1} \wedge |z|\le{R}\} $$ On the boundaries of $\Omega$ we impose dirichlet boundary conditions so basically the problem is the following: \begin{align} -(u_{xx}+u_{yy})=1\quad in\ \Omega\\ u=0\quad on \ \partial\Omega \end{align} Now, I have to find the exact solution of this problem. I decide to procede as follows:
I used polar coordinates (r,$\theta$), therefore my equation turns out to be $$-(u_{rr}+\frac{1}{r}u_{r}+\frac{1}{r^{2}}u_{\theta\theta})=1$$
I made the following assumption: since the data of the problem are radially symmetric, I want to search for a solution which is radially symmetric. Hence, I assume that in polar coordinates, u is a function only of the radius as $$-(u_{rr}+\frac{1}{r}u_{r})=1$$
At this point I can solve it: $$ \frac{1}{r}(ru_{r})_{r}=-1 $$ Integrating we obtain $$ u_{r}=-\frac{r}{2}+\frac{C_{1}}{r}$$ and integrate another time $$ u=-\frac{r^2}{4}+C_{1}\ln(r)+C_{2}$$
Now imposing u(1)=u(R)=0 we get $$C_{1}=\frac{R^{2}-1}{4\ln(R)}$$ and $$C_{2}=1/4$$ and the problem is solved.
Is my procedure entirely correct? Thank you in advance!