# Chain rule in Sobolev space

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.

• Does the fact that Lipschitz functions are differentiable a.e. not help you here? Commented Sep 13, 2022 at 3:16
• @ Daniel Shapero, if $A$ is a null set in R, the set $u^{-1}(A)$ maybe not be a null set. Commented Sep 13, 2022 at 3:47

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