What can we say about the boundary of the level set of a Sobolev function?

Question

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!

Answer 1

I figured out the problem later and until today I could have time to write it down.

The proof of the rectifiability of free boundary $\partial \{u>0\}$ requires the Lipchitz regularity of $u$ across the free boundary and the nondegeneracy of the function $u$, see theorem 3.2-theorem 4.5 in Alt and Caffarelli(1981).

In the proof of the Lipchitz regularity of $u$, the authors proves that if $u(x)>0$, then $|\nabla u(x)|$ is bounded, and then they claim that $u$ is Lipchitz. At first I thought it was a problem, because if $|\partial\{u>0\}|$>0, then how can one give a bound for $|\nabla u(x)|$? That is the reason I asked the question here.

Later I found there was no problem. By Gilbarg and Trudinger, $Du^+=Du$ on $\{u>0\}$ and $Du^+=0$ otherwise. This means the weak derivative of $u$ is always $0$ on $\{u=0\}$ no matter how crazy $|\partial\{u>0\}|$ is. Then by a well known but nontrivial result that $C^{0,1}=W^{1,\infty}$, one can finally say $u$ is Lipchitz.

To tell the truth, I was reluctant to post the answer here, because I think the general question related to the boundary of a level set of a function is still meaningful, even just for the mere object of level set of a "nice" function. I have seen some references studying partial results of level sets of functions. For example, this question is related to the generalization of Morse-Sard Theorem for "nice" functions, coarea formula for $\mathcal{H}^s$ rectifiable sets and so on. My adviser told me the regularity of the free boundary of a solution of a general fully nonlinear equation is not known, and the problem is very hard.

By the way, one can easily construct a Lipschitz function $u$ such that $\partial\{u>0\}$ can be as crazy as possible.

Answer 2

Since $\partial\{u>0\}$ has locally finite $H^{n-1}$ measure $|\partial\{u>0\}\cap B_r|=0$ and hence $|\partial\{u>0\}|=0$.