Lemma 7.13.18736729F21D9

Proof. We first prove the chain rule

for a $C^1$ function $\Phi \colon \mathbb R\to \mathbb R$ with bounded derivative; note that $\Phi (g)\in L^1_{\mathrm{loc}}(D)$, since $|\Phi (g)|\leq |\Phi (0)|+\| \Phi '\| _\infty |g|$. Let $V\Subset D$ be open. The mollifications $g_\varepsilon =g*\rho _\varepsilon $ converge to $g$ in $L^1(V)$, together with their first derivatives $\partial _ ag_\varepsilon =(\partial _ ag)*\rho _\varepsilon $, by the remark preceding lemma 2.9.14; passing to a sequence $\varepsilon _ k\to 0$, we may also assume that $g_{\varepsilon _ k}\to g$ almost everywhere on $V$. Since $\Phi $ is Lipschitz, $\Phi (g_{\varepsilon _ k})\to \Phi (g)$ in $L^1(V)$. Moreover,

tends to zero in $L^1(V)$: the first term is bounded by $\| \Phi '\| _\infty |\partial _ ag_{\varepsilon _ k}-\partial _ ag|$, and the second tends to zero by dominated convergence, since it tends to zero almost everywhere, $\Phi '$ being continuous, and is bounded by $2\| \Phi '\| _\infty |\partial _ ag|$. For $\psi \in C^\infty _ c(V)$ the classical chain rule gives $\int \Phi (g_{\varepsilon _ k})\partial _ a\psi =-\int \Phi '(g_{\varepsilon _ k})\partial _ ag_{\varepsilon _ k}\, \psi $, and letting $k\to \infty $ we obtain $\int \Phi (g)\partial _ a\psi =-\int \Phi '(g)\partial _ ag\, \psi $. As $V$ was arbitrary, this proves the chain rule.

Choose a smooth function $\chi $ on $\mathbb R$ supported in $(-2,2)$, with $0\le \chi \le 1$ and $\chi =1$ on $[-1,1]$, and put $\Phi _\varepsilon (t)=\int _0^ t\chi (s/\varepsilon )\, \mathrm ds$ for $\varepsilon >0$. Then $\Phi _\varepsilon $ is smooth with $0\leq \Phi _\varepsilon '\leq 1$, $|\Phi _\varepsilon |\le 2\varepsilon $, and by the chain rule

as $\varepsilon \to 0$, by dominated convergence, since $\chi (g/\varepsilon )\to \mathbf1_{\{ g=0\} }$ pointwise and $|\chi (g/\varepsilon )\nabla g|\leq |\nabla g|$. On the other hand, $\Phi _\varepsilon (g)\to 0$ uniformly, so $\nabla \Phi _\varepsilon (g)\to 0$ in the sense of distributions. Since convergence in $L^1_{\mathrm{loc}}(D)$ implies convergence in the sense of distributions, the locally integrable function $\mathbf1_{\{ g=0\} }\nabla g$ defines the zero distribution, and hence vanishes almost everywhere (proposition 2.5.3). This is the first assertion.

If $g\in W^{2,1}_{\mathrm{loc}}(D)$, then each $\partial _ ag$ belongs to $W^{1,1}_{\mathrm{loc}}(D)$ and, by the first assertion, vanishes almost everywhere on $\{ g=0\} $. Applying the first assertion to $\partial _ ag$, formed with any measurable representative, shows that $\nabla \partial _ ag=0$ almost everywhere on $\{ \partial _ ag=0\} $, a set which contains $\{ g=0\} $ up to a null set. Since there are finitely many derivatives, the union of the exceptional null sets is null. $\square$