Relaxation of singular functionals defined on Sobolev spaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 5 (2000), pp. 71-85.
@article{COCV_2000__5__71_0,
     author = {Ben Belgacem, Hafedh},
     title = {Relaxation of singular functionals defined on {Sobolev} spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {71--85},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2000},
     zbl = {0936.49008},
     mrnumber = {1745687},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2000__5__71_0/}
}
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Ben Belgacem, Hafedh. Relaxation of singular functionals defined on Sobolev spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 5 (2000), pp. 71-85. http://www.numdam.org/item/COCV_2000__5__71_0/

[1] E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string. J. Elasticity 25 ( 1991) 137-148. | MR 1111364 | Zbl 0734.73094

[2] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 ( 1984) 125-145. | MR 751305 | Zbl 0565.49010

[3] G. Anzellotti, S. Baldo and D. Percivale, Dimension reduction in variational problems, asymptotic development in Γ-convergence, and thin structures in elasticity. Asymptot. Anal 9 ( 1994) 61-100. | MR 1285017 | Zbl 0811.49020

[4] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 ( 1977) 337-403. | MR 475169 | Zbl 0368.73040

[5] J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 ( 1984) 225-253. | MR 759098 | Zbl 0549.46019

[6] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 ( 1987) 13-52. | MR 906132 | Zbl 0629.49020

[7] H. Ben Belgacem, Une méthode de Γ-convergence pour un modèle de membrane non linéaire. C. R. Acad. Sci. Paris. Sér. I Math. ( 1996) 845-849. | MR 1446591 | Zbl 0878.73005

[8] H. Ben Belgacem, Modélisation de structures minces en élasticité non linéaire. Thèse de l'Université Pierre et Marie Curie, Paris ( 1996).

[9] G. Bouchitté, I. Fonseca and J. Malý, The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. Roy. Soc. Edinburgh Sect. A 128( 1998) 463-479. | MR 1632814 | Zbl 0907.49008

[10] P.G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity. North-Holland, Amesterdam ( 1988). | MR 936420 | Zbl 0648.73014

[11] B. Dacorogna, Quasiconvexity and relaxation of non convex problems in the calculus of variations. J. Funct. Anal. 46 ( 1982) 102-118. | MR 654467 | Zbl 0547.49003

[12] B. Dacorogna, Remarques sur les notions de polyconvexité, quasiconvexité et convexité de rang 1. J. Math. Pures Appl. 64 ( 1985) 403-438. | MR 839729 | Zbl 0609.49007

[13] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin, Appl. Math. Sci. 78 ( 1989. | MR 990890 | Zbl 0703.49001

[14] B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectoriel cases. Acta Math. 178 ( 1997) 1-37. | MR 1448710 | Zbl 0901.49027

[15] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Paris ( 1974). | MR 463993 | Zbl 0281.49001

[16] I. Fonseca, The lower quasiconvex envelope of the stored energy for an elastic crystal. J. Math. Pures Appl. 67 ( 1988) 175-195. | MR 949107 | Zbl 0718.73075

[17] I. Fonseca, Variational techniques for problems in materials science. Progr. Nonlinear Differential Equations Appl. 25 ( 1996) 162-175. | MR 1414499 | Zbl 0871.49016

[18] I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent. Ann. Inst. H. Poincaré 14 ( 1997) 309-338. | Numdam | MR 1450951 | Zbl 0868.49011

[19] R.V. Kohn and G. Strang, Explicit relaxation of a variational problem in optimal design. Bull. Amer. Math. Soc. 9 ( 1983) 211-214. | MR 707959 | Zbl 0527.49002

[20] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems I, II and III. Comm. Pure Appl. Math. 39 ( 1986) 113-137, 139-182, 353-377. | MR 820342 | Zbl 0621.49008

[21] H. Le Dret and A. Raoult, Le modèle de membrane non linéaire comme limite variationnelle de l'élasticité non linéaire tridimensionnelle. C. R. Acad. Sci. Paris Sér. I Math. ( 1993) 221-226. | MR 1231426 | Zbl 0781.73037

[22] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of three-dimensional nonlinear elasticity. J. Math. Pures Appl. 74 ( 1995) 549-578. | MR 1365259 | Zbl 0847.73025

[23] P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manuscripta Math. 51 ( 1985) 1-28. | MR 788671 | Zbl 0573.49010

[24] P. Marcellini, On the definition and weak lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré 3 ( 1986) 391-409. | Numdam | MR 868523 | Zbl 0609.49009

[25] C.B. Jr. Morrey, Quasi-convexity and the lower semi-continuity of multiple integrals. Pacific J. Math. 2 ( 1952) 25-53. | MR 54865 | Zbl 0046.10803

[26] C.B. Jr. Morrey, Multiple Integrals in the Calculus of Variations. Springer, Berlin ( 1966). | MR 202511 | Zbl 0142.38701

[27] S. Müller, Variational models for microstructure and phase transitions, to appear in Proc. C.I.M.E. summer school "Calculus of variations and geometrie evolution problems". Cetraro ( 1996). | MR 1731640 | Zbl 0968.74050

[28] R.W. Ogden, Large deformation isotropic elasticity: On the correlation of the theory and experiment for compressible rubberlike solids. Proc. Roy. Soc. London Ser. A 328 ( 1972). | Zbl 0245.73032

[29] E.T. Rockafellar, Convex Analysis. Princeton University Press ( 1970). | MR 274683 | Zbl 0193.18401

[30] L. Tartar, Compensated Compactness and Applications to Partial Differential Equations, in Nonlinear Analysis and Mechanics, Heriot-Watt Symp. Vol. IV, R.J. Knops Ed. Pitman, London ( 1979). | MR 584398 | Zbl 0437.35004

[31] V. Zhikov, Lavrentiev phenomenon and homogenization for some variational problems. C. R. Acad. Sci. Paris Sér. I Math. ( 1993) 435-439. | MR 1209262 | Zbl 0783.35005