AbstractWe 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 [G. Anzellotti,
Pairings between measures and bounded functions and compensated compactness,
Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid,
Divergence-measure fields and hyperbolic conservation laws,
Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], 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 \mathrm{BV}.
We remark that
the standard pairing in general does not share this property.
On the spectral synthesis of bounded functions