Singular points for a free boundary problem

We will analyse the singular points, that is $X^j$ such that $u\notin C^{1,1}(B_r(X^j))$ for any $r>0$, of solutions to $$\mathrm{\Delta }u=-{\chi }_{\{u>0\}}\text{ in }{B}_1.$$ In $\mathbb{R}3$ we will show that the singular set consists of isolated points and a part that is locally contained in a one dimensional $C1$ manifold.

We will also classify the singular points in $\mathbb{R}3$ and show that there are only three kinds of such point and give explicit assymptotics for the solution at such points.