Relaxation of convex functionals : the gap problem

 Access to full text

 Full (PDF)

1. [1] Acerbi E., Maso G.Dal, New lower semicontinuity results for polyconvex integrals, Cal. Var.2 (1994) 329-337. Zbl0810.49014MR1385074
2. [2] Acerbi E., Fusco N., Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal.86 (1984) 125-145. Zbl0565.49010MR751305
3. [3] Ball J., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal.63 (1977) 337-403. Zbl0368.73040MR475169
4. [4] Ball J.M., Murat F., W1,p quasiconvexity and variational problems for multiple integrals, J. Funct. Anal.58 (1984) 225-253. Zbl0549.46019MR759098
5. [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. Zbl0921.49004MR1941478
6. [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. A128 (1998) 463-479. Zbl0907.49008MR1632814
7. [7] Bouchitté G., Fonseca I., Mascarenhas L., A global method for relaxation, Arch. Rational Mech. Anal.144 (1998) 1-46. Zbl0921.49004MR1656477
8. [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. Zbl0735.49008
9. [9] Celada P., Dal Maso G., Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. Henri Poincaré Anal. Non Linéaire11 (1994) 661-691. Zbl0833.49013MR1310627
10. [10] Dacorogna B., Direct Methods in the Calculus of Variations, Applied Math. Sciences, 78, Springer-Verlag, 1989. Zbl0703.49001MR990890
11. [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. Zbl0723.49007MR1071650
12. [12] Dal Maso G., Sbordone C., Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z.218 (1995) 603-609. Zbl0822.49010MR1326990
13. [13] Evans L.C., Gariepy R.F., Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. Zbl0804.28001MR1158660
14. [14] Fan X., Zhao D., A class of De Giorgi type and Hölder continuity, Nonlinear Anal. TMA36 (A) (1999) 295-318. Zbl0927.46022MR1688232
15. [15] Fonseca I., Malý J., Relaxation of multiple integrals below the growth exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire14 (1997) 309-338. Zbl0868.49011MR1450951
16. [16] Fonseca I., Marcellini P., Relaxation of multiple integrals in subcritical Sobolev spaces, J. Geom. Anal.7 (1997) 57-81. Zbl0915.49011MR1630777
17. [17] Fusco N., Hutchinson J.E., A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math.85 (1995) 35-50. Zbl0874.49015MR1329439
18. [18] Gangbo W., On the weak lower semicontinuity of energies with polyconvex integrands, J. Math. Pures Appl.73 (1994) 455-469. Zbl0829.49011MR1300984
19. [19] Ioffe A.D., On lower semicontinuity of integral functionals I, SIAM J. Control Optim.15 (1977) 521-538. Zbl0361.46037MR637234
20. [20] James R.D., Spector S.J., The formation of filamentary voids in solids, J. Mech. Phys. Solids39 (1991) 783-813. Zbl0761.73020MR1120242
21. [21] James R.D., Spector S.J., The formation of filamentary voids in solids, Proc. Roy. Soc. Edinburgh Sect. A123 (1993) 681-691. 
22. [22] J. Malý, Weak lower semicontinuity of polyconvex and quasiconvex integrals, Preprint, Vortragsreihe, Bonn, 1993. Zbl0813.49017MR1237608
23. [23] Malý J., Lower semicontinuity of quasiconvex integrals, Manuscripta Math.85 (1994) 419-428. Zbl0862.49017MR1305752
24. [24] Marcellini P., Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math.51 (1985) 1-28. Zbl0573.49010MR788671
25. [25] Marcellini P., On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire3 (1986) 391-409. Zbl0609.49009MR868523
26. [26] Morrey C.B., Multiple integrals in the Calculus of Variations, Springer, Berlin, 1966. Zbl0142.38701MR202511
27. [27] Müller S., Spector S.J., An existence theory for nonlinear elasticity that allows for cavitation, Arch. Rational Mech. Anal.131 (1995) 1-66. Zbl0836.73025MR1346364
28. [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. Zbl0949.74009MR1811313
29. [29] Zhikov V.V., On Lavrentiev's phenomenon, Russian J. Math. Phys.3 (1995) 249-269. Zbl0910.49020MR1350506
30. [30] Zhikov V.V., On some variational problems, Russian J. Math. Phys.5 (1997) 105-116. Zbl0917.49006MR1486765

1. Irene Fonseca, Nicola Fusco, Paolo Marcellini, Topological degree, Jacobian determinants and relaxation