I wanna discretize the following free boundary problem

Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$.

I apply finite difference method and I need to put some initial value on stencil points around the origin and rotate. Could you help me how can I find reasonable initial values?