On the lower semicontinuity of supremal functionals defined on measures

Michele Gori

Bollettino dell'Unione Matematica Italiana (2006)

  • Volume: 9-B, Issue: 2, page 327-369
  • ISSN: 0392-4033

Abstract

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In this paper we consider two particular classes of supremal functionals defined on Radon measures and we find necessary and sufficient conditions for their lower semicontinuity with respect to the weak* convergence. Some applications to the minimization of functionals defined on BV are presented.

How to cite

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Gori, Michele. "On the lower semicontinuity of supremal functionals defined on measures." Bollettino dell'Unione Matematica Italiana 9-B.2 (2006): 327-369. <http://eudml.org/doc/289617>.

@article{Gori2006,
abstract = {In this paper we consider two particular classes of supremal functionals defined on Radon measures and we find necessary and sufficient conditions for their lower semicontinuity with respect to the weak* convergence. Some applications to the minimization of functionals defined on BV are presented.},
author = {Gori, Michele},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {327-369},
publisher = {Unione Matematica Italiana},
title = {On the lower semicontinuity of supremal functionals defined on measures},
url = {http://eudml.org/doc/289617},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Gori, Michele
TI - On the lower semicontinuity of supremal functionals defined on measures
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/6//
PB - Unione Matematica Italiana
VL - 9-B
IS - 2
SP - 327
EP - 369
AB - In this paper we consider two particular classes of supremal functionals defined on Radon measures and we find necessary and sufficient conditions for their lower semicontinuity with respect to the weak* convergence. Some applications to the minimization of functionals defined on BV are presented.
LA - eng
UR - http://eudml.org/doc/289617
ER -

References

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  1. ACERBI, E. - BUTTAZZO, G. - PRINARI, N., The class of functionals which can be represented by a supremum, J. Convex Analysis, 9 (2002), 225-236. Zbl1012.49010
  2. ALICANDRO, R. - BRAIDES, A. - CICALESE, M., L energies on discontinuous functions, Preprint. 
  3. AMBROSIO, L., Existence theory for a new class of variational problems, Arch. Rational Mech. Anal., 111 (1990), 291-322. Zbl0711.49064
  4. AMBROSIO, L. - BUTTAZZO, G., Weak lower semicontinuous envelope of functions defined on a space of measures, Ann. Mat. pura appl., 150 (1988), 311-340. Zbl0648.49009
  5. AMBROSIO, L. - DAL MASO, G., On the relaxation in B V ( Ω , m ) of quasi-convex integrals, J. Func. Anal., 109 (1992), 76-97. Zbl0769.49009
  6. AMBROSIO, L. - FUSCO, N. - PALLARA, D., Functions of bounded variation and free discontinuity problems, Oxford University Press, Inc. New York, 2000. Zbl0957.49001
  7. ANZELLOTTI, G. - BUTTAZZO, G. - DAL MASO, G., Dirichlet problem for demi-coercive functionals, Nonlinear Anal., 10 (1986), 603-613. Zbl0612.49008
  8. ARONSSON, G. - CRANDALL, M.G. - JUUTINEN, P., A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.), 41 (2004),no. 4, 439-505. Zbl1150.35047
  9. BARRON, E.N. - JENSEN, R.R. - WANG, C.Y., Lower semicontinuity of L functionals, Ann. I. H. Poincaré - AN18 (2001), 495-517. Zbl1034.49008
  10. BARRON, E.N. - LIU, W., Calculus of variations in L , Appl. Math. Optim., 35 (1997), 237-263. Zbl0871.49017
  11. BOUCHITTÉ, G., Représentation intégrale de fonctionelles convexes sur un espace de measures, Ann. Univ. Ferrara, 18 (1987), 113-156. Zbl0721.49041
  12. BOUCHITTÉ, G. - BUTTAZZO, G., New lower semicontinuity results for nonconvex functionals defined on measures, Nonlinear Anal., 15 (1990), 679-692. Zbl0736.49007
  13. BOUCHITTÉ, G. - BUTTAZZO, G., Integral representation of nonconvex functionals defined on measures, Ann. Inst. Henri Poincaré, 9 (1992), 101-117. Zbl0757.49012
  14. BOUCHITTÉ, G. - BUTTAZZO, G., Relaxation for a class of nonconvex functionals defined on measures, Ann. Inst. Henri Poincaré, 10 (1993), 345-361. Zbl0791.49016
  15. BOUCHITTÉ, G. - VALADIER, M., Integral representation of convex functionals on a space of measures, J. funct. Analysis, 80 (1988), 398-420. Zbl0662.46009
  16. BUTTAZZO, G., Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Res. Notes Math. Ser., 207, Longman, Harlow (1989). 
  17. DE GIORGI, E. - AMBROSIO, L., Un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend., 82 (1988), 199-210. 
  18. DE GIORGI, E. - AMBROSIO, L. - BUTTAZZO, G., Integral representation and relaxation for functionals defined on measures, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend., 81 (1987), 7-13. Zbl0713.49018
  19. EVANS, L.C. - GARIEPY, R.F., Measure theory and fine properties of functions, Studies in advanced mathematics, CRC Press, Boca Raton, Florida, 1992. 
  20. GIUSTI, E., Minimal surfaces and functions of bounded variation, Birkhäuser, Basel, 1984. Zbl0545.49018
  21. GOFFMAN, C. - SERRIN, J., Sublinear functions of measure and variational integrals, Duke Math J., 31 (1964), 159-178. Zbl0123.09804
  22. GORI, M., On the definition on BV of a supremal functional, Preprint n. 2.449.1417 (October 2002) Università di Pisa. 
  23. GORI, M. - MAGGI, F., On the lower semicontinuity of supremal functionals, ESAIM: COCV., 9 (2003), 135-143. Zbl1066.49010
  24. GORI, M. - MAGGI, F., The common root of the geometric conditions in Serrin's lower semicontinuity theorem, to appear in Annali Mat. pura appl. Zbl1164.49004
  25. JENSEN, R., Uniqueness of Lipchitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. Zbl0789.35008
  26. ROCKAFELLAR, R.T. - WETS, R.J-B., Variational Analysis, Die Grundlehren der mathematischen Wissenschaften317, Springer-Verlag, Berlin, 1998. 
  27. SERRIN, J., On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139-167. Zbl0102.04601

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