Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

Micol Amar; Virginia De Cicco; Nicola Fusco

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 14, Issue: 3, page 456-477
  • ISSN: 1292-8119

Abstract

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New L1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a representation formula for the relaxed functional is also obtained.

How to cite

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Amar, Micol, De Cicco, Virginia, and Fusco, Nicola. "Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 456-477. <http://eudml.org/doc/90878>.

@article{Amar2007,
abstract = { New L1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a representation formula for the relaxed functional is also obtained. },
author = {Amar, Micol, De Cicco, Virginia, Fusco, Nicola},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Semicontinuity; relaxation; BV functions; capacity; semicontinuity},
language = {eng},
month = {11},
number = {3},
pages = {456-477},
publisher = {EDP Sciences},
title = {Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands},
url = {http://eudml.org/doc/90878},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Amar, Micol
AU - De Cicco, Virginia
AU - Fusco, Nicola
TI - Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/11//
PB - EDP Sciences
VL - 14
IS - 3
SP - 456
EP - 477
AB - New L1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a representation formula for the relaxed functional is also obtained.
LA - eng
KW - Semicontinuity; relaxation; BV functions; capacity; semicontinuity
UR - http://eudml.org/doc/90878
ER -

References

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  1. M. Amar, and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients. Ann. Inst. Henri Poincaré11 (1994) 91–133.  Zbl0842.49016
  2. M. Amar and V. De Cicco, Relaxation in BV for a class of functionals without continuity assumptions. NoDEA Nonlinear Differential Equations Appl. (to appear).  Zbl1153.49016
  3. M. Amar, V. De Cicco and N. Fusco, A relaxation result in BV for integral functionals with discontinuous integrands. ESAIM: COCV13 (2007) 396–412.  Zbl1330.49008
  4. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, New York (2000).  Zbl0957.49001
  5. G. Anzellotti, G. Buttazzo and G. Dal Maso, Dirichlet problem for demi-coercive functionals. Nonlinear Anal.10 (1986) 603–613.  Zbl0612.49008
  6. G. Bouchitté and M. Valadier, Integral representation of convex functionals on a space of measures. J. Funct. Anal.80 (1988) 398–420.  Zbl0662.46009
  7. G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation. Arch. Rat. Mech. Anal.145 (1998) 51–98.  Zbl0921.49004
  8. G. Buttazzo, Semicontinuity, Relaxation and Integral Representation Problems in the Calculus of Variations. Pitman Res. Notes in Math., Longman, Harlow (1989).  
  9. M. Carriero, G. Dal Maso, A. Leaci and E. Pascali, Relaxation of the non-parametric Plateau problem with an obstacle. J. Math. Pures Appl.67 (1988) 359–396.  Zbl0617.49018
  10. G. Dal Maso, Integral representation on B V ( Ω ) of Γ-limits of variational integrals. Manuscripta Math.30 (1980) 387–416.  Zbl0435.49016
  11. G. Dal Maso, On the integral representation of certain local functionals. Ricerche di Matematica32 (1983) 85–113.  Zbl0543.49001
  12. G. Dal Maso, An Introduction to Γ-convergence. Birkhäuser, Boston (1993).  
  13. V. De Cicco and G. Leoni, A chain rule in L 1 ( div ; Ω ) and its applications to lower semicontinuity. Calc. Var. Partial Differential Equations19 (2004) 23–51.  Zbl1056.49019
  14. V. De Cicco, N. Fusco and A. Verde, On L1-lower semicontinuity in B V ( Ω ) . J. Convex Analysis12 (2005) 173–185.  Zbl1115.49011
  15. V. De Cicco, N. Fusco and A. Verde, A chain rule formula in B V ( Ω ) and its applications to lower semicontinuity. Calc. Var. Partial Differential Equations28 (2007) 427–447.  Zbl1136.49011
  16. E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.58 (1975) 842–850.  
  17. E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia3 (1979) 63–101.  
  18. H. Federer and W.P. Ziemer, The Lebesgue set of a function whose distribution derivatives are p-th power summable. Indiana Un. Math. J.22 (1972) 139–158.  Zbl0238.28015
  19. I. Fonseca and G. Leoni, On lower semicontinuity and relaxation. Proc. Royal Soc. Edinb., Sect. A, Math.131 (2001) 519–565.  Zbl1003.49015
  20. I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L1. SIAM J. Math. Anal.23 (1992) 1081–1098.  Zbl0764.49012
  21. I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in BV ( Ω , p ) for integrands f ( x , u , u ) . Arch. Rat. Mech. Anal.123 (1993) 1–49.  Zbl0788.49039
  22. N. Fusco, F. Giannetti and A. Verde, A remark on the L1-lower semicontinuity for integral functionals in BV. Manuscripta Math.112 (2003) 313–323.  Zbl1030.49014
  23. N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem. NoDEA Nonlinear Differential Equations Appl.13 (2006) 425–433.  Zbl1215.49024
  24. M. Gori and F. Maggi, The common root of the geometric conditions in Serrin's semicontinuity theorem. Ann. Mat. Pura Appl.184 (2005) 95–114.  Zbl1164.49004
  25. M. Gori, F. Maggi and P. Marcellini, On some sharp conditions for lower semicontinuity in L1. Diff. Int. Eq.16 (2003) 51–76.  Zbl1028.49012
  26. F. Maggi, On the relaxation on BV of certain non coercive integral functionals. J. Convex Anal.10 (2003) 477–489.  Zbl1084.49015
  27. M. Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani. Ann. Scuola Norm. Sup. Pisa18 (1964) 515–542.  Zbl0152.24402
  28. Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Siberian Math. J.9 (1968) 1039–1045.  Zbl0176.44402

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