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The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. (English) Zbl 0907.49008
The paper deals with the problem of the relaxation with respect to the \(W^{1,p}_{\text{loc}}\) topology of an integral functional \[ F(u,U)=\int_U f(\nabla u) dx \] whose integrand is nonnegative and satisfies a growth condition from above of order \(q\), with \(q\geq p\). In a previous paper by I. Fonseca and J. Malý [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, No. 3, 309-338 (1997; Zbl 0868.49011)] it was proved that the relaxed functional \({\mathcal F}(u,U)\) is, as a function of the open set \(U\subset{\mathbb R}^n\), the trace of a Borel measure. The main result of the paper is the characterization of the absolutely continuous part of this measure (when the measure is a Radon measure) as \(Q(f(\nabla u))\), where \(Qf\) is the quasiconvex envelope of \(f\). This characterization holds whenever \(p>1\) and \(p>q-q/n\).
Reviewer: L.Ambrosio (Pisa)

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
Citations:
Zbl 0868.49011
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References:
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