Homogenization of quadratic complementary energies: a duality example

Hélia Serrano

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 2, page 165-173
  • ISSN: 0862-7959

Abstract

top
We study an example in two dimensions of a sequence of quadratic functionals whose limit energy density, in the sense of Γ -convergence, may be characterized as the dual function of the limit energy density of the sequence of their dual functionals. In this special case, Γ -convergence is indeed stable under the dual operator. If we perturb such quadratic functionals with linear terms this statement is no longer true.

How to cite

top

Serrano, Hélia. "Homogenization of quadratic complementary energies: a duality example." Mathematica Bohemica 136.2 (2011): 165-173. <http://eudml.org/doc/197065>.

@article{Serrano2011,
abstract = {We study an example in two dimensions of a sequence of quadratic functionals whose limit energy density, in the sense of $\Gamma $-convergence, may be characterized as the dual function of the limit energy density of the sequence of their dual functionals. In this special case, $\Gamma $-convergence is indeed stable under the dual operator. If we perturb such quadratic functionals with linear terms this statement is no longer true.},
author = {Serrano, Hélia},
journal = {Mathematica Bohemica},
keywords = {$\Gamma $-convergence; oscillatory behaviour; Young measure; conjugate functional; -convergence; oscillatory behavior; Young measure; conjugate functional},
language = {eng},
number = {2},
pages = {165-173},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Homogenization of quadratic complementary energies: a duality example},
url = {http://eudml.org/doc/197065},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Serrano, Hélia
TI - Homogenization of quadratic complementary energies: a duality example
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 2
SP - 165
EP - 173
AB - We study an example in two dimensions of a sequence of quadratic functionals whose limit energy density, in the sense of $\Gamma $-convergence, may be characterized as the dual function of the limit energy density of the sequence of their dual functionals. In this special case, $\Gamma $-convergence is indeed stable under the dual operator. If we perturb such quadratic functionals with linear terms this statement is no longer true.
LA - eng
KW - $\Gamma $-convergence; oscillatory behaviour; Young measure; conjugate functional; -convergence; oscillatory behavior; Young measure; conjugate functional
UR - http://eudml.org/doc/197065
ER -

References

top
  1. Ball, J., 10.1007/BFb0024945, Lectures Notes in Physics 344. Springer, Berlin (1989). (1989) MR1036070DOI10.1007/BFb0024945
  2. Braides, A., Γ -convergence for Beginners, Oxford University Press, Oxford (2002). (2002) Zbl1198.49001MR1968440
  3. Braides, A., Defranceschi, A., Homogenization of Multiple Integrals, Oxford University Press (1998). (1998) Zbl0911.49010MR1684713
  4. Cioranescu, D., Donato, P., An Introduction to Homogenization, Oxford University Press, Oxford (1999). (1999) Zbl0939.35001MR1765047
  5. Maso, G. Dal, An Introduction to Γ -Convergence, Birkhäuser, Basel (1993). (1993) MR1201152
  6. Giorgi, E. De, Franzoni, T., Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei VIII. Ser, Rend. Cl. Sci. Mat. 58 (1975), Italien 842-850. (1975) Zbl0339.49005MR0448194
  7. Girault, V., Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations, Springer, Berlin (1986). (1986) Zbl0585.65077MR0851383
  8. Jikov, V. V., Kozlov, S. M., Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals, Springer, Berlin (1994). (1994) MR1329546
  9. Pedregal, P., Parametrized Measures and Variational Principles, Birkäuser, Basel (1997). (1997) Zbl0879.49017MR1452107
  10. Pedregal, P., 10.1137/S0036141003425696, SIAM J. Math. Anal. 36 (2004), 423-440. (2004) Zbl1077.49012MR2111784DOI10.1137/S0036141003425696
  11. Pedregal, P., Serrano, H., 10.1016/j.na.2008.09.007, Nonlinear Anal., Theory Methods Appl. 70 (2009), 4178-4189. (2009) MR2514750DOI10.1016/j.na.2008.09.007
  12. Serrano, H., 10.1016/j.jmaa.2009.05.056, J. Math. Anal. Appl. 359 (2009), 311-321. (2009) Zbl1167.49016MR2542177DOI10.1016/j.jmaa.2009.05.056
  13. Young, L. C., Lectures on the Calculus of Variations and Optimal Control Theory, Launders Company, Philadelphia (1980). (1980) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.