Given a nonnegative function, $u$, defined in some domain $D\subset R^n$, denote
$$\Lambda(u)=\{x\in D, u=0\}, \Omega(u)=\{x\in D, u>0\}, F=F(u)=\partial\Lambda\cap\partial N.$$
We are intersted in studying the regularity properties of $F(u)$ at the origin, for $u$ a function belong to $P_1$, where the set of functions $P_r(0<r<\infty)$ is defined as follows:
$u\in P_r(M)$ if
(a) $u\in C^{1,1}(B_r)$, $\sup\limits_{B_r}|D_{ij}u|\leq M$ (for any two directions $i,j$, not necessarily orthogonal),
(b)$ u\geq 0$, $0\in F(u)$,
(c)$ \Delta u=1$ in $\Omega(u).$
The main theorem is , unless $\Lambda(u)$ is very "thin", at the origin, $F(u)$ is, in a neiborhood of it, the graph of a $C^1$ function and inparticular $u$ is $C^2$ on $\overline{\Omega}$.
More precisely, we define the minimum diameter of a (bounded) set $S$ ($m.d.(S)$) as the infimum of the distances between pairs of parallel hyperplanes such that $S$ is containd in the strip determined by them.
we measure the thinness of $\Lambda(u)$ in $B_r$ by the quantity
$$\delta_r(\Lambda)=\frac{m.d.(\Lambda\cap B_r)}{r}.$$
Notice that $$2\geq \delta_r(\Lambda)\geq C\frac{|\Lambda\cap B_r|}{|B_r|}.$$