A nonautonomous chain rule in $W^{1,p}$ and $BV$

In this paper we consider the chain rule formula for compositions $x\mapsto F(x, u(x))$ in the case when $u$ has a Sobolev or BV regularity and $F(x,z)$ is separately Sobolev, or BV, with respect to $x$ and $C^1$ with respect to $z$. Our results extend to this "nonautonomous" case the results known for compositions $x\mapsto F(u(x))$.

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