On the divergence of the composition of irregular fields with BV functions

TL;DR

This paper introduces nonlinear pairing measures for divergence rules in BV functions, enabling characterization of lower semicontinuous functionals and their relaxation in Sobolev spaces.

Contribution

It develops a family of pairing measures depending on jump set representatives, ensuring divergence rule validity and lower semicontinuity in BV and Sobolev spaces.

Findings

Introduces nonlinear pairing measures for divergence of BV functions.

Characterizes pairings that ensure lower semicontinuity of functionals.

Shows these pairings as relaxations of Sobolev space functionals.

Abstract

We introduce a family of (nonlinear) pairing measures that ensure the validity of the divergence rule for composite functions $\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a bounded divergence-measure vector field, and $u$ is a scalar function of bounded variation. The elements of the family depend on the choice of the pointwise representative of $u$ on its jump set. Beyond the standard properties, such as the Coarea and Gauss-Green formulas on sets of finite perimeter, this flexibility allows us to characterize the pairings that ensure the lower semicontinuity of the corresponding functionals along sequences converging in $L^1$ with controlled precise values. We show that these lower semicontinuous pairings arise as the relaxation of integral functionals defined in Sobolev spaces.