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 [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 BV. We remark that the standard pairing in general does not share this property.
Pairings between bounded divergence-measure vector fields and BV functions / Crasta, G.; De Cicco, V.; Malusa, A.. - In: ADVANCES IN CALCULUS OF VARIATIONS. - ISSN 1864-8258. - 15:4(2022), pp. 787-810. [10.1515/acv-2020-0058]
Pairings between bounded divergence-measure vector fields and BV functions
Crasta G.;De Cicco V.;Malusa A.
2022
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 [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 BV. We remark that the standard pairing in general does not share this property.File | Dimensione | Formato | |
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