Calculus of Variations and Geometric Measure Theory

L. Ambrosio - G. Crasta - V. De Cicco - G. De Philippis

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

created by ambrosio on 03 Nov 2011
modified by dephilipp on 30 Oct 2017


Accepted Paper

Inserted: 3 nov 2011
Last Updated: 30 oct 2017

Journal: Manuscripta Math.
Year: 2012


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 $x\mapsto F(u(x))$.

Tags: GeMeThNES
Keywords: Chain Rule, $BV$ functions