The nonlinear membrane model : a Young measure and varifold formulation

Med Lamine Leghmizi; Christian Licht; Gérard Michaille

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 3, page 449-472
  • ISSN: 1292-8119

Abstract

top
We establish two new formulations of the membrane problem by working in the space of W Γ 0 1 , p ( Ω , 𝐑 3 ) -Young measures and W Γ 0 1 , p ( Ω , 𝐑 3 ) -varifolds. The energy functional related to these formulations is obtained as a limit of the 3 d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classical formulation. The second formulation moreover accounts for concentration effects.

How to cite

top

Leghmizi, Med Lamine, Licht, Christian, and Michaille, Gérard. "The nonlinear membrane model : a Young measure and varifold formulation." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2005): 449-472. <http://eudml.org/doc/244820>.

@article{Leghmizi2005,
abstract = {We establish two new formulations of the membrane problem by working in the space of $W^\{1,p\}_\{\Gamma _0\}(\Omega ,\mathbf \{R\}^3)$-Young measures and $W^\{1,p\}_\{\Gamma _0\}(\Omega ,\mathbf \{R\}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the $3d$ formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classical formulation. The second formulation moreover accounts for concentration effects.},
author = {Leghmizi, Med Lamine, Licht, Christian, Michaille, Gérard},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {membrane; Young measures; varifolds; energy functional; variational convergence},
language = {eng},
number = {3},
pages = {449-472},
publisher = {EDP-Sciences},
title = {The nonlinear membrane model : a Young measure and varifold formulation},
url = {http://eudml.org/doc/244820},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Leghmizi, Med Lamine
AU - Licht, Christian
AU - Michaille, Gérard
TI - The nonlinear membrane model : a Young measure and varifold formulation
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 3
SP - 449
EP - 472
AB - We establish two new formulations of the membrane problem by working in the space of $W^{1,p}_{\Gamma _0}(\Omega ,\mathbf {R}^3)$-Young measures and $W^{1,p}_{\Gamma _0}(\Omega ,\mathbf {R}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the $3d$ formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classical formulation. The second formulation moreover accounts for concentration effects.
LA - eng
KW - membrane; Young measures; varifolds; energy functional; variational convergence
UR - http://eudml.org/doc/244820
ER -

References

top
  1. [1] H. Attouch, Variational Convergence for Functions and Operators. Applicable Mathematics Series, Pitman Advanced Publishing Program (1984). Zbl0561.49012MR773850
  2. [2] E.J. Balder, Lectures on Young measures theory and its applications in economics. Workshop di Teoria della Misura e Analisi Reale, Grado, 1997, Rend. Istit. Univ. Trieste 31 Suppl. 1 (2000) 1–69. Zbl1032.91007
  3. [3] K. Bhattacharya and R.D. James, A theory of thin films of martinsitic materials with applications to microactuators. J. Mech. Phys. Solids 47 (1999) 531–576. Zbl0960.74046
  4. [4] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin. Appl. Math. Sciences 78 (1989). Zbl0703.49001MR990890
  5. [5] Dal Maso, An introduction to Γ -convergence. Birkäuser, Boston (1993). Zbl0816.49001MR1201152
  6. [6] 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
  7. [7] L. Freddi and R. Paroni, The energy density of martensitic thin films via dimension reduction. Rapporto di ricerca n 9 / 2003 del dipartimento di Matematica e Informatica dell’Università di Udine. Zbl1072.35185
  8. [8] D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients. Arch. Rational Mech. Anal. 119 (1991) 329–365. Zbl0754.49020
  9. [9] H. Le Dret and A. Raoult, The nonlinear membrane model as Variational limit in nonlinear three-dimensional elasticity. J. Math. Pures Appl., IX. Ser. 74 (1995) 549–578. Zbl0847.73025
  10. [10] P. Pedregal, Parametrized measures and variational Principle. Birkhäuser (1997). Zbl0879.49017MR1452107
  11. [11] M.A. Sychev, A new approach to Young measure theory, relaxation and convergence in energy. Ann. Inst. Henri Poincaé 16 (1999) 773–812. Zbl0943.49012
  12. [12] M. Valadier, Young measures. Methods of Nonconvex Analysis, A. Cellina Ed. Springer-Verlag, Berlin. Lect. Notes Math. 1446 (1990) 152–188. Zbl0738.28004
  13. [13] M. Valadier, A course on Young measures. Workshop di Teoria della Misura e Analisi Reale, Grado, September 19–October 2, 1993, Rend. Istit. Mat. Univ. Trieste 26 Suppl. (1994) 349–394 Zbl0880.49013

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.