Homogenization of variational problems in manifold valued Sobolev spaces

Jean-François Babadjian; Vincent Millot

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 4, page 833-855
  • ISSN: 1292-8119

Abstract

top
Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185–206]. For energies with superlinear or linear growth, a Γ-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation being the object of [Babadjian and Millot, Calc. Var. Part. Diff. Eq.36 (2009) 7–47].

How to cite

top

Babadjian, Jean-François, and Millot, Vincent. "Homogenization of variational problems in manifold valued Sobolev spaces." ESAIM: Control, Optimisation and Calculus of Variations 16.4 (2010): 833-855. <http://eudml.org/doc/250739>.

@article{Babadjian2010,
abstract = { Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185–206]. For energies with superlinear or linear growth, a Γ-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation being the object of [Babadjian and Millot, Calc. Var. Part. Diff. Eq.36 (2009) 7–47]. },
author = {Babadjian, Jean-François, Millot, Vincent},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Homogenization; Γ-convergence; manifold valued maps; -convergence},
language = {eng},
month = {10},
number = {4},
pages = {833-855},
publisher = {EDP Sciences},
title = {Homogenization of variational problems in manifold valued Sobolev spaces},
url = {http://eudml.org/doc/250739},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Babadjian, Jean-François
AU - Millot, Vincent
TI - Homogenization of variational problems in manifold valued Sobolev spaces
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/10//
PB - EDP Sciences
VL - 16
IS - 4
SP - 833
EP - 855
AB - Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185–206]. For energies with superlinear or linear growth, a Γ-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation being the object of [Babadjian and Millot, Calc. Var. Part. Diff. Eq.36 (2009) 7–47].
LA - eng
KW - Homogenization; Γ-convergence; manifold valued maps; -convergence
UR - http://eudml.org/doc/250739
ER -

References

top
  1. R. Alicandro and C. Leone, 3D-2D asymptotic analysis for micromagnetic energies. ESAIM: COCV6 (2001) 489–498.  Zbl0989.35009
  2. L. Ambrosio and G. Dal Maso, On the relaxation in B V ( Ω ; m ) of quasiconvex integrals. J. Funct. Anal.109 (1992) 76–97.  Zbl0769.49009
  3. J.-F. Babadjian and V. Millot, Homogenization of variational problems in manifold valued B V -spaces. Calc. Var. Part. Diff. Eq.36 (2009) 7–47.  Zbl1169.74037
  4. F. Béthuel, The approximation problem for Sobolev maps between two manifolds. Acta Math.167 (1991) 153–206.  Zbl0756.46017
  5. F. Béthuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal.80 (1988) 60–75.  Zbl0657.46027
  6. F. Béthuel, H. Brézis and J.M. Coron, Relaxed energies for harmonic maps, in Variational methods, Paris (1988), H. Berestycki, J.M. Coron and I. Ekeland Eds., Progress in Nonlinear Differential Equations and Their Applications4, Birkhäuser, Boston (1990) 37–52.  
  7. A. Braides, Homogenization of some almost periodic coercive functional. Rend. Accad. Naz. Sci. XL103 (1985) 313–322.  Zbl0582.49014
  8. A. Braides and A. Defranceschi, Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications12. Oxford University Press, New York (1998).  Zbl0911.49010
  9. A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems. Arch. Rational Mech. Anal.135 (1996) 297–356.  Zbl0924.35015
  10. H. Brézis, J.M. Coron and E.H. Lieb, Harmonic maps with defects. Comm. Math. Phys.107 (1986) 649–705.  Zbl0608.58016
  11. B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag (1989).  Zbl0703.49001
  12. B. Dacorogna, I. Fonseca, J. Malý and K. Trivisa, Manifold constrained variational problems. Calc. Var. Part. Diff. Eq.9 (1999) 185–206.  Zbl0935.49006
  13. G. Dal Maso, An Introdution to Γ-convergence. Birkhäuser, Boston (1993).  
  14. I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels. Dunod, Gauthiers-Villars, Paris (1974).  Zbl0281.49001
  15. I. Fonseca and S. Müller, Quasiconvex integrands and lower semicontinuity in L1. SIAM J. Math. Anal.23 (1992) 1081–1098.  Zbl0764.49012
  16. I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in B V ( Ω ; p ) for integrands f ( x , u , u ) . Arch. Rational Mech. Anal.123 (1993) 1–49.  Zbl0788.49039
  17. I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal.29 (1998) 736–756.  Zbl0920.49009
  18. M. Giaquinta, L. Modica and J. Souček, Cartesian currents in the calculus of variations, Modern surveys in Mathematics37-38. Springer-Verlag, Berlin (1998).  
  19. M. Giaquinta, L. Modica and D. Mucci, The relaxed Dirichlet energy of manifold constrained mappings. Adv. Calc. Var.1 (2008) 1–51.  Zbl1157.49043
  20. P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl. (4)117 (1978) 139–152.  Zbl0395.49007
  21. S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rational Mech. Anal.99 (1987) 189–212.  

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.