Chain rule in the Sobolev space $W^{1,p}$

Solution 1:

Assume first that $1\leq p <\infty$

If $u\in C^\infty(\bar{U})$ then clearly $v=F(u)\in C^1(\bar{U})$ and $\nabla v=F'(u)\nabla u$.

Now if $u$ is a general $W^{1,p}$ function then take a sequence $u_k \to u$ in $W^{1,p}$ with $u_k\in C^\infty(\bar{U})$ and such that $u_k\to u$ and $\nabla u_k \to \nabla u$ pointwise a.e. in $U$. Then $$ |F(u)-F(u_k)| \leq M|u-u_k|, $$ with $M=\| F'\|_\infty$. On the other hand we also have $$ F'(u_k(x))\nabla u_k(x) \to F'(u(x))\nabla u(x), \qquad \text{ for a.e. } x\in U, $$ and moreover $| F'(u_k)\nabla u_k|\leq M|\nabla u_k|$. Since $\nabla u_k \to \nabla u$ in $L^p$, by the dominated convegence theorem, $F'(u_k)\nabla u_k \to F'(u)\nabla u$ in $L^p(U)$. Combining this with the first estimate we get that $F(u)\in W^{1,p}(U)$ and $\nabla F(u)= F'(u)\nabla u$.

If $p=\infty$ then we can simply note that $W^{1,\infty}(U)$ is the space of Lipschitz continuous functions in $U$, so take $u$ a Lipschitz function with Lipschitz constant $N$, and $M$ the Lipschitz constant of $F$ as before, then $$ |F(u(x))-F(u(y))|\leq M|u(x)-u(y)| \leq NM|x-y|, \qquad \forall x,\ y\in U. $$ Therefore $F(u)\in W^{1,\infty}(U)$.

Edit: As an extra exercise try to see that the condition $U$ being a $C^1$ domain is not needed when $1\leq p<\infty$ (try to prove that $F(u)\in W^{1,p}(U)$ whenever $u\in C^\infty(U)\cap W^{1,p}(U)$).