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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

##### 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.)
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##### 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
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