# A free boundary problem by finite difference method

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?

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Is this in the plane? Also, this looks like it would have a radially symmetric analytical solution. why do you need a finite difference method for it. –  Paul Tupper Jul 24 '11 at 3:33
Yes, it is in the plane. By potential theory techniques we can prove that $\Omega$ is a ball and then the symmetric analytical solution is determined but $\Omega$ is unknown in advanced. We are looking for a numerical solution of the multiphase version of the problem which has no analytical solution. –  Reza Jul 24 '11 at 17:00