Integration by parts on set $\{u>0\}$

Question

Consider a $C^\infty$ function satisfying $\Delta u=f$ in $\Bbb R^n$. Suppose that the super-level set $\{u>0\}$ is bounded, so $\partial\{u>0\}$ is compact. Can one integrate by parts on this? Namely, if $\partial\{u>0\}$ were smooth, then $$\int_{\{u>0\}} |Du|^2=\int_{\partial\{u>0\}}u \frac{\partial u}{\partial n}-\int_{\{u>0\}}fu=-\int_{\{u>0\}}fu.$$ But $\partial\{u>0\}$ could be something strange. Is this kind of thing ok because $u=0$ on $\partial\{u>0\}$ anyway? I tried showing that $\{u>0\}$ is a set of finite perimeter, but no luck. The literature only seems to have results of this type for when $u$ is harmonic.

Answer 1

The set $\Omega = \{u>0\}$ is exhausted by the sets $\Omega_\epsilon = \{u>\epsilon\}$, for positive $\epsilon>0$. By Sard's lemma, $\partial\Omega_\epsilon $ is smooth for a.e. $\epsilon$. For such $\epsilon$, consider the function $u_\epsilon = u-\epsilon$ on $\Omega_\epsilon$; integration by parts yields $$ \int_{\Omega_\epsilon} |Du_\epsilon|^2 = -\int_{\Omega_\epsilon}fu_\epsilon $$ hence $$ \int_{\Omega_\epsilon} (|Du |^2 +fu) = \epsilon \int_{\Omega_\epsilon}f $$ As long as $\int_\Omega |f| < \infty$, the right hand side tends to zero as $\epsilon\to 0$. Hence $$ \int_{\Omega } (|Du |^2 +fu) = \lim_{\epsilon\to 0}\int_{\Omega_\epsilon} (|Du |^2 +fu) = 0 $$