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

TL;DR

This paper extends the chain rule formula to compositions involving functions with Sobolev or BV regularity and nonautonomous functions that are separately Sobolev or BV in the spatial variable, generalizing known results.

Contribution

It introduces a nonautonomous chain rule formula for compositions where the outer function varies with position and has Sobolev or BV regularity, expanding previous autonomous results.

Findings

Established a chain rule for Sobolev and BV functions with nonautonomous compositions

Generalized known chain rule results to more complex, position-dependent functions

Extended the applicability of chain rule formulas in analysis of PDEs and variational problems

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