Abstract

We study the lower semicontinuous envelope of a class of functionals with linear growth defined on mappings from the n-dimensional ball Bn into RN\mathbb{R}^N, that are constrained to take values into a smooth submanifold Y\mathcal{Y} of RN\mathbb{R}^N

Author information

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

Mariano Giaquinta

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy

giaquinta@sns.it

Domenico Mucci

Dip. di Matematica, Università di Parma, Viale G. P. Usberti 53/A, 43100 Parma, Italy

domenico.mucci@unipr.it

References

  1. E. Acerbi, G. Dal Maso: New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differ. Equ. 2 (1994) 329–372.https://doi.org/10.1007/BF01235534Google Scholar
  2. L. Ambrosio: Metric space valued functions of bounded variation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17 (1990) 439–478.Google Scholar
  3. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  4. F. Bethuel: The approximation problem for Sobolev maps between manifolds, Acta Math. 167 (1992) 153–206.https://doi.org/10.1007/BF02392449Google Scholar
  5. F. Bethuel, J.-M. Coron, F. Demengel, F. Helein: A cohomological criterium for density of smooth maps in Sobolev spaces between two manifolds, in: Nematics, Mathematical and Physical Aspects, J.-M. Coron et al. (ed.), NATO ASI Series, Series C 332, Kluwer, Dordrecht (1991) 15–23.https://doi.org/10.1007/978-94-011-3428-6_2Google Scholar
  6. G. Bouchitté, G. Buttazzo: New lower semicontinuity results for nonconvex functionals defined on measures, Nonlinear Anal., Theory Methods Appl. 15 (1990) 679–692.https://doi.org/10.1016/0362-546X(90)90007-4Google Scholar
  7. H. Federer: Geometric Measure Theory, Grundl. Math. Wissenschaft. 153, Springer, Berlin (1969).Google Scholar
  8. M. Giaquinta, G. Modica, J. Souček: Cartesian currents in the calculus of variations, I, II, Ergebnisse Math. Grenzgebiete (III. Ser.) 37, 38, Springer, Berlin (1998).https://doi.org/10.1007/978-3-662-06218-0Google Scholar
  9. M. Giaquinta, G. Modica, J. Souček: Variational problems for maps of bounded variation with values in S1, Calc. Var. Partial Differ. Equ. 1 (1993) 87–121.https://doi.org/10.1007/BF02163266Google Scholar
  10. M. Giaquinta, D. Mucci: Maps into manifolds and currents: area and W 1,2 -, W 1/2 -, BV - energies, Edizioni della Normale, CRM Series, Sc. Norm. Super. Pisa, Pisa (2006).Google Scholar
  11. M. Giaquinta, D. Mucci: On sequences of maps into a manifold with equibounded W 1/2 - energies, J. Funct. Anal. 225 (2005) 94–146.https://doi.org/10.1016/j.jfa.2005.02.013Google Scholar
  12. M. Giaquinta, D. Mucci: The BV -energy of maps into a manifold: relaxation and density results, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 5 (2006) 483–548.https://doi.org/10.2422/2036-2145.2006.4.04Google Scholar
  13. M. Giaquinta, D. Mucci: Erratum and addendum to: The BV -energy of maps into a manifold: relaxation and density results, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6 (2007) 185–194.https://doi.org/10.2422/2036-2145.2007.1.09Google Scholar
  14. M. Giaquinta, D. Mucci: The Dirichlet energy of mappings from B 3 into a manifold: density results and gap phenomenon, Calc. Var. Partial Differ. Equ. 20 (2004) 367–397.https://doi.org/10.1007/s00526-003-0225-xGoogle Scholar
  15. Y. G. Reshetnyak: Weak convergence of completely additive vector functions on a set, Siberian Math. J. 9 (1968) 1039–1045.https://doi.org/10.1007/BF02196453Google Scholar
  16. L. Simon: Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. National Univ., Centre Math. Anal. 3, Austral. National Univ., Canberra (1983).Google Scholar
  17. B. White: Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988) 1–17.https://doi.org/10.1007/BF02392271Google Scholar

Suggested citation

M. Giaquinta, D. Mucci. “Relaxation Results for a Class of Functionals with Linear Growth Defined on Manifold Constrained Mappings.” Journal of Convex Analysis 15 (2008), No. 4, 719–751. https://doi.org/10.68381/jca15049

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.