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.  
  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).  
  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.  
  4. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford University Press, New York (2000).  
  5. G. Anzellotti, G. Buttazzo and G. Dal Maso, Dirichlet problem for demi-coercive functionals. Nonlinear Anal.10 (1986) 603–613.  
  6. G. Bouchitté and M. Valadier, Integral representation of convex functionals on a space of measures. J. Funct. Anal.80 (1988) 398–420.  
  7. G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation. Arch. Rat. Mech. Anal.145 (1998) 51–98.  
  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.  
  10. G. Dal Maso, Integral representation on B V ( Ω ) of Γ-limits of variational integrals. Manuscripta Math.30 (1980) 387–416.  
  11. G. Dal Maso, On the integral representation of certain local functionals. Ricerche di Matematica32 (1983) 85–113.  
  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.  
  14. V. De Cicco, N. Fusco and A. Verde, On L1-lower semicontinuity in B V ( Ω ) . J. Convex Analysis12 (2005) 173–185.  
  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.  
  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.  
  19. I. Fonseca and G. Leoni, On lower semicontinuity and relaxation. Proc. Royal Soc. Edinb., Sect. A, Math.131 (2001) 519–565.  
  20. I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L1. SIAM J. Math. Anal.23 (1992) 1081–1098.  
  21. I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in BV ( Ω , ℝ p ) for integrands f ( x , u , ∇ u ) . Arch. Rat. Mech. Anal.123 (1993) 1–49.  
  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.  
  23. N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem. NoDEA Nonlinear Differential Equations Appl.13 (2006) 425–433.  
  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.  
  25. M. Gori, F. Maggi and P. Marcellini, On some sharp conditions for lower semicontinuity in L1. Diff. Int. Eq.16 (2003) 51–76.  
  26. F. Maggi, On the relaxation on BV of certain non coercive integral functionals. J. Convex Anal.10 (2003) 477–489.  
  27. M. Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani. Ann. Scuola Norm. Sup. Pisa18 (1964) 515–542.  
  28. Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Siberian Math. J.9 (1968) 1039–1045.  

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