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

Ambrosio Luigi - Crasta Graziano - De Cicco Virginia - De Philippis Guido

year: 2012
journal: Manuscripta Math.
abstract: 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 $x\mapsto F(u(x))$.

The paper is available on the cvgmt preprint server.