Anzellotti, Gabriele Pairings between measures and bounded functions and compensated compactness. (English) Zbl 0572.46023 Ann. Mat. Pura Appl., IV. Ser. 135, 293-318 (1983). 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 Cited in 4 ReviewsCited in 237 Documents 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 PDFBibTeX XMLCite \textit{G. Anzellotti}, Ann. Mat. Pura Appl. (4) 135, 293--318 (1983; Zbl 0572.46023) Full Text: DOI References: [1] G.Anzellotti,On the existence of the rates of stress and displacement for Prandtl-Reuss plasticity, Quaterly of Appl. Math., July 1983. · Zbl 0521.73030 [2] G.Anzellotti,On the extremal stress and displacement in Hencky plasticity, Duke Math. J., March 1984. · Zbl 0548.73022 [3] G.Anzellotti,On the minima of functionals with linear growth, to appear. · Zbl 0589.49027 [4] Anzellotti, G.; Giaquinta, M., Funzioni BV e tracce, Rend. Sem. Mat. Padova, 60, 1-21 (1978) · Zbl 0432.46031 [5] H.Federer,Geometric measure theory, Springer-Verlag (1969). · Zbl 0176.00801 [6] Gagliardo, E., Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Padova, 27, 284-305 (1957) · Zbl 0087.10902 [7] Giusti, E., Minimal surfaces and functions of bounded variation, Notes on Pure Math (1977), Canberra: Australian National University, Canberra · Zbl 0402.49033 [8] R.Kohn - R.Temam,Dual spaces of stresses and strains, with application to Hencky plasticity, to appear. · Zbl 0532.73039 [9] Miranda, M., Superfici cartesiane generalizzate ed insiemi di perimetro finito sui prodotti cartesiani, Ann. Scuola Normale Sup. Pisa, S. III, 18, 513-542 (1964) · Zbl 0152.24402 [10] Murat, F., Compacité par compensation, Ann. Scuola Normale Sup. Pisa, 5, IV (1978) · Zbl 0399.46022 [11] L.Schwartz,Théorie des distributions, Hermann (1957, 1959). · Zbl 0089.09601 [12] L.Tartar,The compensated compactness method applied to systems of conservation laws, in « Systems of Nonlinear partial differential equations », J. M. Ball (ed.), Reidel Publishing Co. (1983). [13] Temam, R., Navier-Stokes Equation (1977), Amsterdam: North Holland, Amsterdam · Zbl 0383.35057 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.