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Pairings between measures and bounded functions and compensated compactness. (English) Zbl 0572.46023
This 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

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.)
Full Text: DOI
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