In the theory of Sobolev space, we have the following chain rule:


* For a uniformly Lipschitz function $F : \mathbf{R}\to \mathbf{R}$ such that $F(0)=0$,
and $u\in W^{1,1}(\mathbf{R}^n)$, then we have the following chain rule:
$\partial\_j F(u)=F'(u)\circ \partial\_ju$.


But how to define the function $F'(u)$? It seems that we can't define $F'(u(.))$ a.e., and it may be not a measurable function.



Chain rule in Sobolev space
The main difficulty in the proof of the rule is to prove that $\nabla u=0$ a.e. on the set $u^{-1}(\Sigma)$, where $\Sigma$ is the set where $F$ is not differentiable; and where $\nabla u=0$ one defines the product $F'(u)\nabla u$ to be 0, irrespective of the fact that $F'(u)$ is defined or not a such points. See e.g. Leoni, Morini: JEMS 9 pp 219-252


