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:

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