Abstract

A new lower semicontinuity theorem for integral functionals defined on the space BV of functions of bounded variation is proved. This result is obtained by replacing the continuity of the energy density f with respect to the space variable x with a weak differentiability assumption. The proof is based on a new integration by parts formula in the space BV which does not seem to be contained in any of the similar results known in the literature and which could be of independent interest

Author information

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

Virginia De Cicco

Università di Roma, Dip. di Metodi e Modelli Matematici per le Scienze Applicate, Via Scarpa 16, 00161 Roma, Italy

decicco@dmmm.unirom1.it

Nicola Fusco

Università di Napoli, Dipartimento di Matematica e Applicazioni, Via Cinzia, 80126 Napoli, Italy

n.fusco@unina.it

Anna Verde

Università di Napoli, Dipartimento di Matematica e Applicazioni, Via Cinzia, 80126 Napoli, Italy

anverde@unina.it

References

  1. L. Ambrosio: New lower semicontinuity results for integral functionals, Rend. Accad. Naz. Sci. XL 1 (1987) 1–42.Google Scholar
  2. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  3. G. Bouchitté, G. Dal Maso: Integral representation and relaxation of convex local functionals on BV (Ω), Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20 (1993) 483–533.Google Scholar
  4. A. Braides, V. De Cicco: New lower semicontinuity and relaxation results for functionals defined on BV (Ω), Adv. Math. Sci. Appl. 6 (1996) 1–30.Google Scholar
  5. G. Dal Maso: Integral representation on BV (Ω) of Γ-limits of variational integrals, Manusc. Math. 30 (1980) 387–416.https://doi.org/10.1007/BF01301259Google Scholar
  6. V. De Cicco: A lower semicontinuity result for functionals defined on BV (Ω), Ric. Mat. 39 (1990) 293–325.Google Scholar
  7. V. De Cicco: Lower semicontinuity for certain integral functionals on BV (Ω), Boll. U.M.I. 5-B (1991) 291–313.Google Scholar
  8. V. De Cicco, G. Leoni: A chain rule in L1 (div; Ω) and its applications to lower semicontinuity, Calc. Var. Partial Differ. Equ. 19(1) (2004) 23–51.https://doi.org/10.1007/s00526-003-0192-2Google Scholar
  9. E. De Giorgi: Teoremi di Semicontinuità nel Calcolo delle Variazioni, Istituto Nazionale di Alta Matematica (1968-1969).Google Scholar
  10. E. De Giorgi, G. Buttazzo, G. Dal Maso: On the lower semicontinuity of certain integral functionals, Atti Accad. Naz. Lincei, Cl. Sc. Fis. Mat. Natur. Rend. 74 (1983) 274–282.Google Scholar
  11. H. Federer: Geometric Measure Theory, Springer (1969).Google Scholar
  12. I. Fonseca, G. Leoni: Some remarks on lower semicontinuity, Indiana Univ. Math. J. 49 (2000) 617–635.https://doi.org/10.1512/iumj.2000.49.1791Google Scholar
  13. I. Fonseca, G. Leoni: On lower semicontinuity and relaxation, Proc. R. Soc. Edin., Sect. A Math. 131 (2001) 519–565.https://doi.org/10.1017/S0308210501000245Google Scholar
  14. N. Fusco: Dualità e semicontinuità per integrali del tipo dell’area, Rend. Accad. Sci. Fis. Mat. 46 (1979) 81–90.Google Scholar
  15. N. Fusco, F. Giannetti, A. Verde: A remark on the L1 -lower semicontinuity for integral functionals in BV, Manusc. Math. 112 (2003) 313–323.https://doi.org/10.1007/s00229-003-0400-6Google Scholar
  16. A. Gavioli: Necessary and sufficient conditions for the lower semicontinuity of certain integral functionals, Ann. Univ. Ferrara, Sez. VII Sc. Mat. 34 (1988) 219–236.https://doi.org/10.1007/BF02824983Google Scholar
  17. M. Gori, P. Marcellini: An extension of the Serrin’s lower semicontinuity theorem, J. Convex Anal. 9 (2002) 475–502.https://doi.org/10.68381/jca09028Google Scholar
  18. M. Gori, F. Maggi, P. Marcellini: On some sharp conditions for lower semicontinuity in L1, Diff. Int. Eq. 16 (2003) 51–76.https://doi.org/10.57262/die/1356060696Google Scholar
  19. C. Y. Pauc: La Méthode Métrique en Calcul des Variations, Hermann, Paris (1941).Google Scholar
  20. J. Serrin: On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 161 (1961) 139–167.https://doi.org/10.1090/S0002-9947-1961-0138018-9Google Scholar

Suggested citation

V. De Cicco, N. Fusco, A. Verde. “On L^1-Lower Semicontinuity in BV.” Journal of Convex Analysis 12 (2005), No. 1, 173–185. https://doi.org/10.68381/jca12012

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2005. Rights now held by Banach Press.