Abstract

We study the lower semicontinuity properties of autonomous variational integrals whose energy densities are controlled by N-functions.

Author information

Contact details are reproduced from the original publication and may be historical.

Matteo Focardi

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Elvira Mascolo

Dip. di Matematica, Università di Firenze, Viale Morgagni 87/a, 50134 Firenze, Italy

Keywords

  • Quasi-convexity
  • lower semicontinuity
  • Orlicz-Sobolev spaces

Mathematics Subject Classification

49J45

References

  1. E. Acerbi, G. Dal Maso: New lower semicontinuity results for polyconvex integrals case, Calc. Var. 2 (1994) 329–372.https://doi.org/10.1007/BF01235534Google Scholar
  2. E. Acerbi, N. Fusco: Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal. 86 (1984) 125–145.https://doi.org/10.1007/BF00275731Google Scholar
  3. R. A. Adams: Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
  4. J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977) 337–403.https://doi.org/10.1007/BF00279992Google Scholar
  5. P. Celada, G. Dal Maso: Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. H. Poincaré (Anal. non Linéaire) 11 (1995) 661–691.https://doi.org/10.1016/s0294-1449(16)30173-1Google Scholar
  6. T. Bhattacharya, F. Leonetti: A new Poincaré inequality and its applications to the regularity of minimizers of integrals functionals with non-standard growth, Nonlinear Anal. 17 (1991) 833–839.https://doi.org/10.1016/0362-546X(91)90157-VGoogle Scholar
  7. B. Dacorogna: Direct Methods in the Calculus of Variations, Appl. Math Sci. 78, Springer Verlag, Heidelberg and New York, 1989.https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  8. B. Dacorogna, P. Marcellini: Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C. R. Acad. Sci. Paris 311(I) (1990) 393–396.Google Scholar
  9. A. Dall’Aglio, E. Mascolo, G. Papi: Regularity for local minima of functionals with nonstandard growth conditions, Rend. Mat. 18(VII), Roma (1998) 305–326.Google Scholar
  10. G. Dal Maso, C. Sbordone: Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z. 218 (1995) 603–609.https://doi.org/10.1007/BF02571927Google Scholar
  11. E. De Giorgi: Teoremi di semicontinuità nel calcolo delle variazioni, INdAM, Roma, 1968-69.Google Scholar
  12. T. K. Donaldson: Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces, J. Differential Equations 10 (1971) 507–528.https://doi.org/10.1016/0022-0396(71)90009-XGoogle Scholar
  13. T. K. Donaldson, N. S. Trudinger: Orlicz-Sobolev spaces and embedding theorems, J. Funct. Anal. 8 (1971) 42–75.https://doi.org/10.1016/0022-1236(71)90018-8Google Scholar
  14. D. E. Edmunds, B. Opic, L. Pick: Poincaré and Frederichs inequalities in abstract Sobolev spaces, Math. Proc. Camb. Phil. Soc. 113 (1993) 355–379.https://doi.org/10.1017/S0305004100076027Google Scholar
  15. M. Focardi: Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend. Ist. Mat. Univ. Trieste 29 (1997) 141–161.Google Scholar
  16. I. Fonseca, J. Malý: Relaxation of multiple integrals in Sobolev spaces below the growth exponent for the energy density, Ann. Inst. H. Poincaré (Analyse non Linéaire) 14 (1997) 309–338.https://doi.org/10.1016/s0294-1449(97)80139-4Google Scholar
  17. I. Fonseca, P. Marcellini: Relaxation of multiple integrals in subcritical Sobolev spaces, J. Geom. Anal. 7 (1997) 57–81.https://doi.org/10.1007/BF02921705Google Scholar
  18. I. Fonseca, S. Müller: Quasi-convex integrands and lower semicontinuity in L1, SIAM J. Math. Anal. 23 (1992) 1081–98.https://doi.org/10.1137/0523060Google Scholar
  19. A. Fougères: Théorèmes de trace et de prolongement dans les espaces de Sobolev et de Sobolev-Orlicz, C. R. Acad. Sci. Paris 274(A) (1972) 181–184.Google Scholar
  20. N. Fusco: Quasi-convessità e semicontinuità per integrali multipli di ordine superiore, Ricerche di Mat. 29 (1980) 307–323.Google Scholar
  21. N. Fusco, J. E. Hutchinson: A direct proof of lower semicontinuity for polyconvex functionals, Manuscripta Mat. 87 (1995) 35–50.https://doi.org/10.1007/BF02570460Google Scholar
  22. D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer Verlag, New York, 1983.Google Scholar
  23. E. Giusti: Metodi Diretti nel Calcolo delle Variazioni, U.M.I., Bologna, 1994.Google Scholar
  24. J. P. Gossez: Nonlinear elliptic problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math Soc. 55 (1974) 163–205.https://doi.org/10.1090/S0002-9947-1974-0342854-2Google Scholar
  25. J. P. Gossez: Some approximation properties in Orlicz-Sobolev spaces, Studia Mat. 74 (1982) 17–24.https://doi.org/10.4064/sm-74-1-17-24Google Scholar
  26. J. P. Gossez: A remark on strongly nonlinear elliptic bondary value problems, Bol. Soc. Brasil. Mat. 8 (1977) 53–63.https://doi.org/10.1007/BF02584750Google Scholar
  27. M. A. Krasnosel’skij, Ya. B. Rutickii: Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.Google Scholar
  28. J. Kristensen: Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand, Proc. Roy. Soc. Edinburgh Sect A 127 (1997) 797–817.https://doi.org/10.1017/S0308210500023830Google Scholar
  29. J. Malý: Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc. Edinburgh Sect A 123 (1993) 681–691.https://doi.org/10.1017/S0308210500030900Google Scholar
  30. J. Malý: Weak lower semicontinuity of polyconvex and quasi-convex integrals, Vortragsreihe 1993, Bonn.https://doi.org/10.1017/S0308210500030900Google Scholar
  31. J. Malý: Lower semicontinuity of quasi-convex integrals, Manus. Math. 85 (1994) 419–428.https://doi.org/10.1007/BF02568208Google Scholar
  32. P. Marcellini: Approximation of quasi-convex functions and lower semicontinuity of multiple integrals, Manus. Math. 51 (1985) 1–28.https://doi.org/10.1007/BF01168345Google Scholar
  33. P. Marcellini: On the definition and the lower semicontinuity of certain quasi-convex integrals, Ann. Inst. H. Poincaré (Analyse non Linéaire) 3 (1986) 391–409.https://doi.org/10.1016/s0294-1449(16)30379-1Google Scholar
  34. P. Marcellini: Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993) 296–333.https://doi.org/10.1006/jdeq.1993.1091Google Scholar
  35. N. G. Meyers: Quasi-convexity and the semicontinuity of multiple variational integrals of any order, Trans. Amer. Math Soc. 119 (1965) 125–149.https://doi.org/10.1090/S0002-9947-1965-0188838-3Google Scholar
  36. L. Mini: Rilassamento per Funzionali del Calcolo delle Variazioni, Tesi di Laurea, Università di Firenze (1997).Google Scholar
  37. C. B. Morrey: Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952) 25–53.https://doi.org/10.2140/pjm.1952.2.25Google Scholar
  38. M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces, Pure and Appl. Math., Marcel Dekker, New York, 1981.Google Scholar
  39. N. S. Trudinger: On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473–483.https://doi.org/10.1512/iumj.1968.17.17028Google Scholar
  40. K. Zhang: A construction of quasi-convex functions with linear growth at infinity, Ann. Scuola Norm. Sup. Pisa 19 (1992) 313–326.Google Scholar
  41. W. Ziemer: Weakly Differentiable Functions, GTM 120, Springer Verlag, 1989.https://doi.org/10.1007/978-1-4612-1015-3Google Scholar

Suggested citation

M. Focardi, E. Mascolo. “Lower Semicontinuity of Quasi-Convex Functionals with Non-Standard Growth.” Journal of Convex Analysis 8 (2001), No. 2, 327–347. https://doi.org/10.68381/jca08016

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2001. Rights now held by Banach Press.