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

1. Problem Statement

The paper addresses the fundamental challenge of establishing a chain rule formula for the distributional divergence of a composite function $v(x) = B(x, u(x))$, where:

• $u \in BV(\Omega) \cap L^\infty(\Omega)$ is a scalar function of bounded variation.
• $B: \Omega \times \mathbb{R} \to \mathbb{R}^N$ is a family of bounded vector fields with divergence-measure properties (i.e., $B(\cdot, t) \in DM^\infty(\Omega)$).

In the classical smooth setting ($u, B \in C^1$), the chain rule is:
$$\text{div}(B(x, u)) = (\text{div}_x B)(x, u) + \partial_t B(x, u) \cdot \nabla u$$
However, when $u$ is a BV function, $\nabla u$ is a measure, and the term $\partial_t B(x, u) \cdot \nabla u$ is ill-defined in the classical sense. The core problem is to define a pairing measure between the field $b(x, u) := \partial_t B(x, u)$ and the distributional derivative $Du$ that:

1. Extends the chain rule to the measure-theoretic sense.
2. Preserves classical properties (Coarea, Gauss-Green formulas).
3. Ensures lower semicontinuity of associated integral functionals under specific convergence modes.
4. Allows for the relaxation of functionals defined on Sobolev spaces to the BV setting.

2. Methodology and Framework

Assumptions on the Vector Field

The authors consider a family of vector fields $b(x, t)$ satisfying:

• Boundedness: $b \in L^\infty(\Omega \times \mathbb{R})$.
• Continuity in $t$: $t \mapsto b(x, t)$ is continuous for a.e. $x$.
• Divergence-Measure: For every $t$, $b(\cdot, t) \in DM^\infty(\Omega)$.
• Uniform Control: The least upper bound of the total variations of the divergences, $\sigma := \bigvee_{t \in \mathbb{R}} |\text{div}_x b(\cdot, t)|$, is a finite Radon measure.

The Concept of Pairing

The divergence of the composite function is decomposed as:
$$\text{div}(B(x, u)) = (\text{div}_x B)(x, u) + (b(x, u), Du)$$
The term $(b(x, u), Du)$ is the pairing measure. The paper introduces two distinct families of pairings depending on the choice of the representative of $u$ on its jump set $J_u$:

1. External Nonlinear $\lambda$-pairing: Defined via a convex combination of the values of the Radon-Nikodym derivative $F(x, t)$ at the approximate limits $u^-$ and $u^+$:
   $$[b(\cdot, u), Du]_\lambda := -\left( (1-\lambda)F(x, u^-) + \lambda F(x, u^+) \right)\sigma + \text{div}(B(x, u))$$
2. Nonlinear $\lambda$-pairing: Defined by evaluating $F$ at a specific intermediate representative $u_\lambda = (1-\lambda)u^- + \lambda u^+$:
   $$(b(\cdot, u), Du)_\lambda := -F(x, u_\lambda)\sigma + \text{div}(B(x, u))$$

Here, $\lambda: \Omega \to [0, 1]$ is a Borel function determining the selection of the representative on the jump set.

3. Key Contributions and Results

A. Generalized Chain Rule and Representation

The authors prove that both families of pairings are well-defined finite Radon measures absolutely continuous with respect to $|Du|$. They provide explicit density representations:

• On the Jump Set ($J_u$): The density depends on the weak normal traces of the field $b$ on $J_u$ and the choice of $\lambda$.
• On the Diffuse Part ($D^d u$): The density is independent of $\lambda$ and is characterized using the Coarea formula. Specifically, for the diffuse part, the density coincides with the weak normal trace of the field $b_t$ on the reduced boundary of the level sets $\{u > t\}$.

B. Generalized Gauss-Green Formulas

The paper establishes a new, very general Gauss-Green formula for sets of finite perimeter $E \subset \Omega$. For both pairing families, the formula relates the integral of the divergence over $E$ (and its boundary) to the normal traces of the composite field. This extends previous results from linear pairings and standard nonlinear pairings to the general nonlinear setting with measure divergence.

C. Lower Semicontinuity (The Main Innovation)

A critical finding is that the standard pairing (corresponding to $\lambda = 1/2$) and the external pairings generally fail to ensure lower semicontinuity of the associated functionals.

• The authors identify a specific class of selections $\lambda \in LM[u]$ such that the pairing $(b(\cdot, u), Du)_\lambda$ becomes lower semicontinuous with respect to sequences converging in $L^1$ with controlled precise values (the $A(u)$ class).
• This optimal pairing is defined using the function $pF(x, a, b) = \max_{t \in [a, b]} F(x, t)$. The lower semicontinuous pairing is:
  $$(b(\cdot, u), Du)_L := -pF(x, u^-, u^+)\sigma + \text{div}(B(x, u))$$
• They also characterize a pairing $(b(\cdot, u), Du)_V$ that ensures the lower semicontinuity of the total variation of the measure.

D. Relaxation of Integral Functionals

The paper proves that the lower semicontinuous pairing arises naturally as the relaxation of integral functionals defined on Sobolev spaces.
Let $F_\phi(u) = \int_\Omega \phi b(x, u) \cdot \nabla u \, dx$ for $u \in W^{1,1}$. The relaxation of this functional with respect to $L^1$ convergence (restricted to sequences satisfying the PR-property) is exactly:
$$\overline{F}_\phi(u) = \int_\Omega \phi \, d(b(x, u), Du)_L$$
This result generalizes previous work (e.g., [28]) which required the divergence measure to have an $L^1$ density, extending it to the general case where the divergence is a singular measure.

4. Significance and Applications

• Unification: The framework unifies linear pairings (where $B(x, t) = tA(x)$) and nonlinear pairings, showing that external pairings are a specific subclass of nonlinear pairings under certain monotonicity conditions.
• Variational Calculus: The identification of the lower semicontinuous pairing is crucial for the calculus of variations in $BV$. It allows for the rigorous formulation of minimization problems involving discontinuous fluxes, plasticity, and free discontinuity problems where the energy density depends on the gradient in a non-smooth way.
• PDEs: The results provide the necessary tools for the weak formulation of PDEs with discontinuous coefficients or fluxes, such as conservation laws ($u_t + \text{div} B(x, u) = 0$) and equations involving the 1-Laplace operator.
• Relaxation Theory: By characterizing the relaxation of functionals from $W^{1,1}$ to $BV$ without assuming absolute continuity of the divergence measure, the paper removes a significant technical barrier in the analysis of irregular fields.

Conclusion

This paper fundamentally advances the theory of divergence-measure fields by introducing a flexible family of nonlinear pairings. It resolves the issue of lower semicontinuity by identifying a specific "optimal" pairing based on the maximum of the Radon-Nikodym derivative over the jump interval. This leads to a complete relaxation formula for integral functionals in the $BV$ setting, extending the applicability of these tools to a much broader class of irregular vector fields than previously possible.