Suppose cylinder and Dirichet's conditions. Are $u(x,y)=\frac{Q}{4} \left(R^2-x^2-y^2\right)$ in $\mathbb R^2$ and $u(x,y,z)=\frac{Q}{6} \left(R^2-x^2-y^2-z^2\right)$ in $\mathbb R^3$ the only solutions?
D.C. are:
$$\begin{cases}
-\nabla u = \rho & \text{when A is an inner point} \\
u=0 & \text{when on the border of } \partial A \\
\end{cases}$$
How to prove
$$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$$
