Bouchitté, Guy; Fonseca, Irene; Malý, Jan The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. (English) Zbl 0907.49008 Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 3, 463-479 (1998). 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) Cited in 26 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation Keywords:lower semicontinuity; quasiconvexity; relaxation; integral functional Citations:Zbl 0868.49011 PDFBibTeX XMLCite \textit{G. Bouchitté} et al., Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 3, 463--479 (1998; Zbl 0907.49008) Full Text: DOI References: [1] DOI: 10.1016/S0294-1449(97)80139-4 · Zbl 0868.49011 [2] Dacorogna, Direct Methods in the Calculus of Variations 78 (1989) · Zbl 0703.49001 [3] DOI: 10.1007/BF02571927 · Zbl 0822.49010 [4] Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (1993) · Zbl 0821.42001 [5] Celada, Ann. Inst. H. Poincaré, Anal. Non Linéaire 11 pp 661– (1994) · Zbl 0833.49013 [6] Carbone, Ricerche Mat. 39 pp 99– (1990) [7] DOI: 10.1016/0022-1236(84)90041-7 · Zbl 0549.46019 [8] DOI: 10.1007/BF00279992 · Zbl 0368.73040 [9] DOI: 10.1007/BF00275731 · Zbl 0565.49010 [10] DOI: 10.1007/BF01235534 · Zbl 0810.49014 [11] DOI: 10.1007/BF02568208 · Zbl 0862.49017 [12] Malý, Proc. Roy. Soc. Edinburgh Sect. A 123 pp 681– (1993) · Zbl 0813.49017 [13] Hajlasz, Ann. Inst. H. Poincare, Anal. Non Lineaire 12 pp 415– (1995) · Zbl 0910.49025 [14] Gangbo, J. Math. Pures Appl. 73 pp 455– (1994) [15] DOI: 10.1007/BF02570460 · Zbl 0874.49015 [16] Morrey, Multiple integrals in the Calculus of Variations (1966) · Zbl 0142.38701 [17] Marcellini, Ann. Inst. H. Poincaré, Anal. Non Linéaire 3 pp 391– (1986) · Zbl 0609.49009 [18] DOI: 10.1007/BF01168345 · Zbl 0573.49010 [19] Dacorogna, C. R. Acad. Sci. Paris Sér. I Math. 311 pp 393– (1990) [20] Federer, Geometric Measure Theory (1996) · Zbl 0874.49001 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.