I'm a beginner of the area of free boundary problem. Let me first give some background:

$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph. Consider the convex set $$K:＝\{v \in L^1_{loc}(\Omega): \nabla v \in L^2(\Omega) \,, v=u^0 \mbox{on $\partial \Omega$}\},$$ where $u^0\ge0,u^0 \in L^1_{loc}(\Omega)$, and $\nabla u^0 \in L^2(\Omega)$.

We are looking for the minimizer $u$ of the functional $$J(v):=\int_{\Omega}(|\nabla v|^2+\chi_{\{v>0\}})$$ in the class $K$.

It is proved that the minimizer $u$ exists and satisfying the following properties: $$u \ge 0 , \, \Delta u=0 \, \mbox{on the open set $\{u>0\}$}, \, \mbox{and $u$ is subharmonic},$$ see section 1-2 in the paper by Alt and Caffarelli here

It is also proved in section 3-5 that $\partial\{u>0\}$ has locally finite $\mathcal{H}^{n-1}$ measure. However, a lot of intermediate theorems such as Corollary 3.3 and Remark 4.2 are based on the fact that $|\partial\{u>0\}|=0$, that is, $|\partial\{u=0\}|=0$. This fact is not proved in the paper, and generally it is not true if $u$ is merely continuous.

Now my question is, why is $|\partial\{u=0\}|=0$ true? I've been stucked on it for a couple of days.

Another related question is, what conditions on a general function $u$, which is not necessarily the minimum of the functional $J$, can guarantee that $|\partial\{u=0\}|=0$? Is the assumption that $u$ is a Sobolev function enough? How about $u$ is subharmonic?

Any suggestions would be appreciated. Thanks!