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arxiv: 2604.14033 · v1 · submitted 2026-04-15 · 🧮 math.FA · math.AP

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

Graziano Crasta , Virginia De Cicco , Annalisa Malusa This is my paper

Pith reviewed 2026-05-10 12:06 UTC · model grok-4.3

classification 🧮 math.FA math.AP
keywords divergence-measure fieldsbounded variationpairing measurescompositionlower semicontinuityrelaxationGauss-Green formulaCoarea formula
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The pith

A family of nonlinear pairing measures lets the divergence rule hold for B(x,u(x)) when B is a bounded divergence-measure field and u is BV.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a family of nonlinear pairing measures to define the divergence of composite functions B(x, u(x)), where B(·, t) is a bounded divergence-measure vector field and u is a scalar function of bounded variation. These measures depend on the choice of pointwise representative of u on its jump set, which provides flexibility to also satisfy the Coarea formula, Gauss-Green formula on sets of finite perimeter, and lower semicontinuity of associated integral functionals under L1 convergence with controlled precise values. The authors show that the lower semicontinuous pairings in this family arise precisely as the relaxation of integral functionals originally defined on Sobolev spaces. A sympathetic reader cares because this extends the classical divergence theorem and integration-by-parts formulas to settings with discontinuous fields and functions that commonly appear in variational problems and PDEs with irregular data.

Core claim

We introduce a family of (nonlinear) pairing measures that ensure the validity of the divergence rule for composite functions B(x,u(x)), where B(·,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 L1 with controlled precise values. We show that these lower semicontinuous pairings arise as

What carries the argument

Nonlinear pairing measures between a bounded divergence-measure vector field B(·,t) and a BV function u, parameterized by the choice of representative of u on its jump set.

Load-bearing premise

Suitable pointwise representatives of u on its jump set can be chosen so that the same pairing simultaneously satisfies the divergence rule, the Coarea formula, the Gauss-Green formula, and the lower-semicontinuity characterization.

What would settle it

An explicit bounded divergence-measure field B and BV function u for which no choice of representative on the jump set produces a pairing that both satisfies the divergence equality and makes the associated functional lower semicontinuous under L1 convergence.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a family of nonlinear pairing measures between bounded divergence-measure vector fields B and scalar BV functions u. The elements of the family are indexed by the choice of pointwise representative of u on its jump set J_u. These pairings are asserted to satisfy the divergence rule for the composition B(x, u(x)), the Coarea and Gauss-Green formulas on sets of finite perimeter, and to characterize lower semicontinuity of associated integral functionals; the lower-semicontinuous pairings are shown to arise as the relaxation of functionals originally defined on Sobolev spaces.

Significance. If the existence of representatives satisfying all properties simultaneously is established, the work would provide a flexible extension of pairing theory in BV spaces to irregular coefficients, with direct implications for relaxation and lower-semicontinuity results in the calculus of variations. The parameterization by representatives on the jump set is a distinctive feature that could enable finer control in applications involving discontinuous fields.

major comments (1)
  1. [Section 3 (definition of the family and main existence statement)] The central claim requires the existence of a single choice of representative of u on J_u that simultaneously satisfies the divergence rule for B(x,u(x)), the Coarea and Gauss-Green formulas, and the lower-semicontinuity characterization used for the relaxation result. The manuscript should supply an explicit joint selection argument (rather than separate verifications for each property) to guarantee that the family is non-empty for general B and u; without it the relaxation theorem rests on an unverified assumption.
minor comments (2)
  1. [Notation and preliminaries] Clarify in the notation section how the different elements of the family are distinguished when multiple representatives are possible.
  2. [Introduction] The abstract states that the pairings 'arise as the relaxation' but does not name the precise functional being relaxed; a one-sentence reminder in the introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying this subtle but important point concerning the simultaneous satisfaction of multiple properties by a single representative. We address the concern directly below.

read point-by-point responses

  1. Referee: [Section 3 (definition of the family and main existence statement)] The central claim requires the existence of a single choice of representative of u on J_u that simultaneously satisfies the divergence rule for B(x,u(x)), the Coarea and Gauss-Green formulas, and the lower-semicontinuity characterization used for the relaxation result. The manuscript should supply an explicit joint selection argument (rather than separate verifications for each property) to guarantee that the family is non-empty for general B and u; without it the relaxation theorem rests on an unverified assumption.

    Authors: We agree that separate verifications do not by themselves guarantee a common representative satisfying all required properties at once, and that an explicit joint selection argument is needed to confirm the family is non-empty. In the present framework the divergence rule and the Coarea/Gauss-Green formulas hold for every choice of representative on J_u (the pairing measure is constructed so that values on an H^{n-1}-negligible subset of J_u do not affect the integrals). The lower-semicontinuity characterization, however, singles out a particular representative. Because any two representatives differ only on an H^{n-1}-null set, and because the divergence and integral identities are unaffected by such null-set modifications, the representative that realizes lower semicontinuity automatically satisfies the remaining properties. We will insert a short lemma in Section 3 that makes this joint selection explicit (first select the representative enforcing the lower-semicontinuity condition, then note that it works for the other identities up to null-set adjustment). revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a family of nonlinear pairing measures indexed by choices of pointwise representatives of the BV function u on its jump set J_u. It then uses this family to establish the divergence rule for B(x,u(x)), Coarea/Gauss-Green formulas, and lower-semicontinuity of relaxed functionals, showing the selected pairings arise as relaxations of Sobolev-space integral functionals. No quoted step reduces a claimed result to a definition or fit of the same quantity by construction, nor does any load-bearing premise collapse to a self-citation chain or imported uniqueness theorem. The derivation remains self-contained within the standard framework of divergence-measure fields and BV theory, with the representative choice providing parametric flexibility rather than tautological equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of suitable pointwise representatives for BV functions on jump sets and on the standard Coarea and Gauss-Green formulas for sets of finite perimeter; no free parameters or new invented entities beyond the pairing family itself are mentioned.

axioms (1)
  • domain assumption Coarea and Gauss-Green formulas hold on sets of finite perimeter for BV functions
    Invoked to ensure the pairings satisfy the standard properties listed in the abstract.
invented entities (1)
  • family of nonlinear pairing measures no independent evidence
    purpose: To ensure the divergence rule holds for B(x,u(x)) and to characterize lower semicontinuity of the associated functionals
    The family is introduced in the paper and depends on the choice of representative of u on its jump set; no independent evidence outside the paper is provided.

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