Pairings between measures and bounded functions and compensated compactness.

*(English)*Zbl 0572.46023This paper deals with the pairings between measures and bounded measurable functions. When \(\mu =Du\) with \(u\in BV(\Omega),\) \(\psi \in L^{\infty}(\Omega,{\mathbb{R}}^ n)\) such that div \(\psi\) is bounded measurable on an open bounded set \(\Omega\) in \({\mathbb{R}}^ n\), then the author develops several properties of the pairing \(<\psi,u>\) and \(<\psi,Du>\). The author obtains a formula of integral representation for \(<\psi,u>\), shows that \(<\psi,Du>\) is a Radon measure on \(\Omega\), absolutely continuous with respect to the measure \(| Du|\) on \(\Omega\) and establishes the relation (Green formula) between the measure \(<\psi,Du>\) and the function \(<\psi,\nu >\) where \(\nu\) (x) denotes the outward unit normal to \(\partial \Omega\). In section 2, the author is concerned with the representation of the density \(\theta\) (\(\psi\),Du) of the measure \(<\psi,Du>\) with respect to the measure \(| Du|\). Other properties of the function \(\theta\) (\(\psi\),Du) are developed. In section 3, the author studies the pairing \(<\psi,\mu >\) when \(\mu\) is a measure whose curl is also a measure and presents some properties of \(<\psi,\mu >\) analogously as in section 1 and section 2. Finally a sequential continuity result of the bilinear mapping \((\psi,\mu)\mapsto <\psi,\mu >\) is established in M(\(\Omega)\).

Reviewer: Ch.Castaing

##### MSC:

46E27 | Spaces of measures |

28A33 | Spaces of measures, convergence of measures |

46A20 | Duality theory for topological vector spaces |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

##### Keywords:

pairings between measures and bounded measurable functions; integral representation; Radon measure; Green formula; sequential continuity
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\textit{G. Anzellotti}, Ann. Mat. Pura Appl. (4) 135, 293--318 (1983; Zbl 0572.46023)

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##### References:

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