The nonlinear membrane model : a Young measure and varifold formulation
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, p. 449-472

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 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.

DOI : https://doi.org/10.1051/cocv:2005014
Classification:  74K15,  35B05,  49J45
Keywords: membrane, Young measures, varifolds
@article{COCV_2005__11_3_449_0,
     author = {Leghmizi, Med Lamine and Licht, Christian and Michaille, G\'erard},
     title = {The nonlinear membrane model : a Young measure and varifold formulation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {11},
     number = {3},
     year = {2005},
     pages = {449-472},
     doi = {10.1051/cocv:2005014},
     zbl = {1081.74027},
     mrnumber = {2148853},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2005__11_3_449_0}
}
Leghmizi, Med Lamine; Licht, Christian; Michaille, Gérard. The nonlinear membrane model : a Young measure and varifold formulation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 449-472. doi : 10.1051/cocv:2005014. http://www.numdam.org/item/COCV_2005__11_3_449_0/

[1] H. Attouch, Variational Convergence for Functions and Operators. Applicable Mathematics Series, Pitman Advanced Publishing Program (1984). | MR 773850 | Zbl 0561.49012

[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. | Zbl 1032.91007

[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. | Zbl 0960.74046

[4] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin. Appl. Math. Sciences 78 (1989). | MR 990890 | Zbl 0703.49001

[5] Dal Maso, An introduction to Γ-convergence. Birkäuser, Boston (1993). | MR 1201152 | Zbl 0816.49001

[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. | Zbl 0920.49009

[7] L. Freddi and R. Paroni, The energy density of martensitic thin films via dimension reduction | Zbl 1072.35185

[8] D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients. Arch. Rational Mech. Anal. 119 (1991) 329-365. | Zbl 0754.49020

[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. | Zbl 0847.73025

[10] P. Pedregal, Parametrized measures and variational Principle. Birkhäuser (1997). | MR 1452107 | Zbl 0879.49017

[11] M.A. Sychev, A new approach to Young measure theory, relaxation and convergence in energy. Ann. Inst. Henri Poincaé 16 (1999) 773-812. | Numdam | Zbl 0943.49012

[12] M. Valadier, Young measures. Methods of Nonconvex Analysis, A. Cellina Ed. Springer-Verlag, Berlin. Lect. Notes Math. 1446 (1990) 152-188. | Zbl 0738.28004

[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 | Zbl 0880.49013