An extension of the pairing theory between divergence-measure fields and BV functions

Abstract

We introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [6,10], all the main properties and features (e.g. coarea, Leibniz and Gauss-Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in BV . We remark that the standard pairing in general does not share this property.

Capacities and 1-strict subsets in metric spaces

“…In Example 6.22 we show that the term C a Du j (Ω) involving the jump part of the variation measure of u is necessary. Very recently, essentially the same result was proved in Euclidean spaces in [13,Proposition 7.3], based on an earlier result [12,Theorem 3.3]. In Euclidean spaces the term C a Du j (Ω) is not needed, but for us the existence of this term makes it necessary to use rather different techniques in the proof, as we will discuss in Remark 6.23.…”

“…Remark 6.23. Recall that in the Euclidean setting, the term C a Du j (Ω) is not needed (see [13,Proposition 7.3]). Having lim i→∞ Ω g w i dµ = Du (Ω) is in fact used in [13] to prove the pointwise convergence, whereas in our setting it seems necessary to construct the approximations "by hand", which makes the proof of Theorem 1.2 rather technically involved.…”

Rigidity and trace properties of divergence-measure vector fields

“…Conversely, one can easily check that given V satisfying (16)- (18), the vector field η defined as in (19) satisfies (12)- (15). Proof.…”

“…Since then, the class DM ∞ (Ω) of divergence-measure, bounded vector fields has been widely studied in view of applications to hyperbolic systems of conservation laws [7,9,10], to continuum and fluid mechanics [8], and to minimal surfaces [28,34] among many others. In particular, the weak normal trace has been studied in different directions, see for instance [1,11,12,13,15,14]. Despite some explicit characterizations of the weak normal trace are available (see the discussion in Section 2.3), a crucial issue coming with this distributional notion is that it is not possible to recover the pointwise value of such a trace by a standard, measure-theoretic limit, see Example 2.7.…”

Anzellotti's pairing theory and the Gauss–Green theorem

“…Finally, in some lower semicontinuity problems for integral functionals defined in Sobolev spaces and in BV , the vector fields with measure-derivative occurred as natural dependence of the integrand with respect to the spatial variable (see [8,21,24]). To this end, we address the reader to the forthcoming paper [19], where the authors introduce a nonlinear version of the pairing suitable for applications to semicontinuity problems.…”