Relaxation of convex functionals : the gap problem
Acerbi, E. ; Bouchitté, G. ; Fonseca, I.
Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003), p. 359-390 / Harvested from Numdam
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@article{AIHPC_2003__20_3_359_0,
     author = {Acerbi, Emilio and Bouchitte, Guy and Fonseca, I.},
     title = {Relaxation of convex functionals : the gap problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {20},
     year = {2003},
     pages = {359-390},
     doi = {10.1016/S0294-1449(02)00017-3},
     mrnumber = {1972867},
     zbl = {1025.49012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2003__20_3_359_0}
}

Acerbi, E.; Bouchitté, G.; Fonseca, I. Relaxation of convex functionals : the gap problem. Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003) pp. 359-390. doi : 10.1016/S0294-1449(02)00017-3. http://gdmltest.u-ga.fr/item/AIHPC_2003__20_3_359_0/

[1] Acerbi E., Maso G.Dal, New lower semicontinuity results for polyconvex integrals, Cal. Var. 2 (1994) 329-337. | MR 1385074 | Zbl 0810.49014

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

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

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

[5] G. Bouchitté, I. Fonseca, G. Leoni, L. Mascarenhas, A global method for relaxation in W1,p and in SBVp, To appear in Arch. Rational Mech. Anal. | MR 1941478 | Zbl 0921.49004

[6] Bouchitté G., Fonseca I., Malý J., 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

[7] Bouchitté G., Fonseca I., Mascarenhas L., A global method for relaxation, Arch. Rational Mech. Anal. 144 (1998) 1-46. | MR 1656477 | Zbl 0921.49004

[8] Carbone L., De Arcangelis R., Further results on Γ-convergence and lower semicontinuity of integral functionals depending on vector-valued functions, Ricerche Mat. 39 (1990) 99-129. | Zbl 0735.49008

[9] Celada P., Dal Maso G., Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. Henri Poincaré Anal. Non Linéaire 11 (1994) 661-691. | Numdam | MR 1310627 | Zbl 0833.49013

[10] Dacorogna B., Direct Methods in the Calculus of Variations, Applied Math. Sciences, 78, Springer-Verlag, 1989. | MR 990890 | Zbl 0703.49001

[11] Dacorogna B., Marcellini P., Semicontinuité pour des intégrandes polyconvexes sans continuité des determinants, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 393-396. | MR 1071650 | Zbl 0723.49007

[12] Dal Maso G., Sbordone C., Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z. 218 (1995) 603-609. | MR 1326990 | Zbl 0822.49010

[13] Evans L.C., Gariepy R.F., Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. | MR 1158660 | Zbl 0804.28001

[14] Fan X., Zhao D., A class of De Giorgi type and Hölder continuity, Nonlinear Anal. TMA 36 (A) (1999) 295-318. | MR 1688232 | Zbl 0927.46022

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

[16] Fonseca I., Marcellini P., Relaxation of multiple integrals in subcritical Sobolev spaces, J. Geom. Anal. 7 (1997) 57-81. | MR 1630777 | Zbl 0915.49011

[17] Fusco N., Hutchinson J.E., A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math. 85 (1995) 35-50. | MR 1329439 | Zbl 0874.49015

[18] Gangbo W., On the weak lower semicontinuity of energies with polyconvex integrands, J. Math. Pures Appl. 73 (1994) 455-469. | MR 1300984 | Zbl 0829.49011

[19] Ioffe A.D., On lower semicontinuity of integral functionals I, SIAM J. Control Optim. 15 (1977) 521-538. | MR 637234 | Zbl 0361.46037

[20] James R.D., Spector S.J., The formation of filamentary voids in solids, J. Mech. Phys. Solids 39 (1991) 783-813. | MR 1120242 | Zbl 0761.73020

[21] James R.D., Spector S.J., The formation of filamentary voids in solids, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 681-691.

[22] J. Malý, Weak lower semicontinuity of polyconvex and quasiconvex integrals, Preprint, Vortragsreihe, Bonn, 1993. | MR 1237608

[23] Malý J., Lower semicontinuity of quasiconvex integrals, Manuscripta Math. 85 (1994) 419-428. | MR 1305752 | Zbl 0862.49017

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

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

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

[27] Müller S., Spector S.J., An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rational Mech. Anal. 131 (1995) 1-66. | MR 1346364 | Zbl 0836.73025

[28] Sivaloganathan J., On cavitation and degenerate cavitation under internal hydrostatic pressure, Roy. Soc. London Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999) 3645-3664. | MR 1811313 | Zbl 0949.74009

[29] Zhikov V.V., On Lavrentiev's phenomenon, Russian J. Math. Phys. 3 (1995) 249-269. | MR 1350506 | Zbl 0910.49020

[30] Zhikov V.V., On some variational problems, Russian J. Math. Phys. 5 (1997) 105-116. | MR 1486765 | Zbl 0917.49006