Gradient of a Lipschitz function on a Riemannian manifold. I'm referring to the article of D. Fischer-Colbrie and R. Schoen The structure of complete stable minimal surfaces in 3-Manifolds of non-negative scalar curvature (journal link, pdf). 
In the proof of theorem 2 the smooth function in the definition (8) is composed with the distance function $r$ (from $0$). This composition is a Lipschitz function (since $r$ is a Lipschitz function) with compact support. Then this composition belongs in the Sobolev space $H^1$ and then it admits a weak gradient.
Now I don't understand because the norm of this gradient is equal to the norm of the derivative of the function in (8).(this fact is implicitly used in the estimates below (8)). 
The distance function $r$ does not lie in $H^1$, then I don't think that I can use the formula of the derivation of a composition (moreover we have to be careful to treat about the gradient of the distance function $r$).
 A: You are right to be suspicious of the chain rule in the context of weak derivatives, but here it is not a problem because $\zeta$ is smooth. Since the distance function $r$ is Lipschitz, it is differentiable at almost every point (Rademacher's theorem) and at all such points the classical chain rule applies. 
Alternatively, you can argue as follows: by construction $\zeta$ is $(3/R)$-Lipschitz, i.e., it satisfies the Lipschitz condition with the constant $3/R$. Also, $r$ is $1$-Lipschitz by the triangle inequality. It is immediate that the composition of $L_1$-Lipschitz function with an $L_2$-Lipschitz function is $L_1L_2$-Lipschitz; this works in all metric spaces. Thus, $\zeta\circ r$ is $(3/R)$-Lipschitz, which implies that its  gradient is bounded by $3/R$. Notice that only an upper bound on $|\nabla (\zeta\circ r)|$ is needed. 
Lipschitz functions are great: their classical (pointwise) gradient exists a.e., and represents the weak gradient (i.e., the fundamental theorem of calculus applies). Buy a Lipschitz function today. [/spam]
