Hello,


as I'm not an analyst, I'm having difficulties with the following, certainly well-known problem: one is given the PDE $\Delta u(x,y)=\sqrt{x^2+y^2}$ in the "region" $x^2+y^2\leq1$ with the boundary coundition $u(x,y)=0$ whenever $x^2+y^2=1$. The most obvious "answer" would be $u(x,y)=\sqrt{x^2+y^2}$, but the partial derivatives of $u(x,y)$ are not defined at $(x,y)=(0,0)$ (the singularity w.r.t. polar coordinates). Am I overlooking something, i.e. is there a well-behaved solution ? 


Any help would be greatly appreciated !


Kind regards,


Stephan.



Poisson equation in the plane
The usual way to approach this would be to first search for radial solutions, i.e. look for functions $v$ such that $v(|(x,y)|)=u(x,y)$ and $v$ solves the equation. This should reduce the problem to an ODE in this special case, from which it is easy to find a solution. 


