Γ-convergence of functionals on divergence-free fields

Nadia Ansini; Adriana Garroni

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

  • Volume: 13, Issue: 4, page 809-828
  • ISSN: 1292-8119

Abstract

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We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ-convergence. We prove that the Γ-limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Γ-limit is also stable under volume constraint and various type of boundary conditions.

How to cite

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Ansini, Nadia, and Garroni, Adriana. "Γ-convergence of functionals on divergence-free fields." ESAIM: Control, Optimisation and Calculus of Variations 13.4 (2007): 809-828. <http://eudml.org/doc/250011>.

@article{Ansini2007,
abstract = { We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ-convergence. We prove that the Γ-limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Γ-limit is also stable under volume constraint and various type of boundary conditions. },
author = {Ansini, Nadia, Garroni, Adriana},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {$\{\cal A\}$-quasiconvexity; divergence-free fields; Γ-convergence; homogenization; -convergence; functionals on divergence free matrix-valued functions; -quasiconvexity; volume constraints},
language = {eng},
month = {9},
number = {4},
pages = {809-828},
publisher = {EDP Sciences},
title = {Γ-convergence of functionals on divergence-free fields},
url = {http://eudml.org/doc/250011},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Ansini, Nadia
AU - Garroni, Adriana
TI - Γ-convergence of functionals on divergence-free fields
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/9//
PB - EDP Sciences
VL - 13
IS - 4
SP - 809
EP - 828
AB - We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ-convergence. We prove that the Γ-limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Γ-limit is also stable under volume constraint and various type of boundary conditions.
LA - eng
KW - ${\cal A}$-quasiconvexity; divergence-free fields; Γ-convergence; homogenization; -convergence; functionals on divergence free matrix-valued functions; -quasiconvexity; volume constraints
UR - http://eudml.org/doc/250011
ER -

References

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  1. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal.86 (1984) 125–145.  
  2. R.A. Adams, Sobolev spaces. Academic Press, New York (1975).  
  3. A. Braides, Γ-convergence for Beginners. Oxford University Press, Oxford (2002).  
  4. A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998).  
  5. A. Braides, I. Fonseca and G. Leoni, A-Quasiconvexity: Relaxation and Homogenization. ESAIM: COCV5 (2000) 539–577.  
  6. G. Dal Maso, An Introduction to Γ -convergence. Birkhäuser, Boston (1993).  
  7. I. Fonseca and S. Müller, A-Quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal.30 (1999) 1355–1390.  
  8. I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradient. SIAM J. Math. Anal.29 (1998) 736–756.  
  9. F. Murat, Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci.8 (1981) 68–102.  
  10. P. Pedregal, Parametrized measures and variational principles. Birkhäuser, Baston (1997).  
  11. P. Suquet, Overall potentials and extremal surfaces of power law or ideally plastic composites. J. Mech. Phys. Solids41 (1993) 981–1002.  
  12. D.R.S. Talbot and J.R. Willis, Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proc. Roy. Soc. London A447 (1994) 365–384.  
  13. D.R.S. Talbot and J.R. Willis, Upper and lower bounds for the overall properties of a nonlinear composite dielectric. II. Periodic microgeometry. Proc. Roy. Soc. London A447 (1994) 385–396.  
  14. L. Tartar, Compensated compactness and applications to partial differential equations. Nonlinerar Analysis and Mechanics: Heriot-Watt Symposium, R. Knops Ed., Longman, Harlow. Pitman Res. Notes Math. Ser.39 (1979) 136–212.  
  15. R. Temam, Navier-Stokes Equations. Elsevier Science Publishers, Amsterdam (1977).  

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