Equi-integrability results for 3D-2D dimension reduction problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002) , pp. 443-470.

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla }_{\alpha }{u}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_{3}{u}_{\epsilon }\right)$ bounded in ${L}^{p}\left(\Omega ;{ℝ}^{9}\right),\phantom{\rule{4pt}{0ex}}1 Here it is shown that, up to a subsequence, ${u}_{\epsilon }$ may be decomposed as ${w}_{\epsilon }+{z}_{\epsilon },$ where ${z}_{\epsilon }$ carries all the concentration effects, i.e. $\left\{{\left|\left({\nabla }_{\alpha }{w}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_{3}{w}_{\epsilon }\right)\right|}^{p}\right\}$ is equi-integrable, and ${w}_{\epsilon }$ captures the oscillatory behavior, i.e. ${z}_{\epsilon }\to 0$ in measure. In addition, if $\left\{{u}_{\epsilon }\right\}$ is a recovering sequence then ${z}_{\epsilon }={z}_{\epsilon }\left({x}_{\alpha }\right)$ nearby $\partial \Omega .$

DOI : https://doi.org/10.1051/cocv:2002063
Classification : 49J45,  74B20,  74G10,  74K15,  74K35
Mots clés : equi-integrability, dimension reduction, lower semicontinuity, maximal function, oscillations, concentrations, quasiconvexity
@article{COCV_2002__7__443_0,
author = {Bocea, Marian and Fonseca, Irene},
title = {Equi-integrability results for 3D-2D dimension reduction problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {443--470},
publisher = {EDP-Sciences},
volume = {7},
year = {2002},
doi = {10.1051/cocv:2002063},
zbl = {1044.49010},
mrnumber = {1925037},
language = {en},
url = {http://www.numdam.org/item/COCV_2002__7__443_0/}
}
Bocea, Marian; Fonseca, Irene. Equi-integrability results for 3D-2D dimension reduction problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002) , pp. 443-470. doi : 10.1051/cocv:2002063. http://www.numdam.org/item/COCV_2002__7__443_0/

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational. Mech. Anal. 86 (1984) 125-145. | MR 751305 | Zbl 0565.49010

[2] E. Acerbi and N. Fusco, An approximation lemma for ${W}^{1,p}$ functions, in Material Instabilities in Continuum Mechanics and Related Mathematical Problems, edited by J.M. Ball. Heriot-Watt University, Oxford (1988). | Zbl 0644.46026

[3] E. Anzelotti, S. Baldo and D. Percivale, Dimensional reduction in variational problems, asymptotic developments in $\Gamma$-convergence, and thin structures in elasticity. Asymptot. Anal. 9 (1994) 61-100. | MR 1285017 | Zbl 0811.49020

[4] E.J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984) 570-598. | MR 747970 | Zbl 0549.49005

[5] J.M. Ball, A version of the fundamental theorem for Young mesures, in PDE's and Continuum Models of Phase Transitions, edited by M. Rascle, D. Serre and M. Slemrod. Springer-Verlag, Berlin, Lecture Notes in Phys. 344 (1989) 207-215. | Zbl 0991.49500

[6] H. Berliocchi and J.-M. Lasry, Intégrands normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129-184. | Numdam | MR 344980 | Zbl 0282.49041

[7] K. Bhattacharya and A. Braides, Thin films with many small cracks. Preprint (2000). | MR 1898090 | Zbl 1011.74042

[8] K. Bhattacharya, I. Fonseca and G. Francfort, An asymptotic study of the debonding of thin films. Arch. Rational. Mech. Anal. 161 (2002) 205-229. | MR 1894591 | Zbl 0999.74079

[9] K. Bhattacharya and R.D. James, A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47 (1999) 531-576. | MR 1675215 | Zbl 0960.74046

[10] A. Braides, Private communication.

[11] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49 (2000) 1367-1404. | MR 1836533 | Zbl 0987.35020

[12] A. Braides and I. Fonseca, Brittle thin films. Appl. Math. Optim. 44 (2001) 299-323. | MR 1851742 | Zbl 0999.49012

[13] S. Conti, I. Fonseca and G. Leoni, $A$ $\Gamma$-convergence result for the two-gradient theory of phase transitions, Preprint 01-CNA-008. Center for Nonlinear Analysis, Carnegie Mellon University (2001). Comm. Pure Applied Math. (to appear). | MR 1894158 | Zbl 1029.49040

[14] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag (1989). | MR 990890 | Zbl 0703.49001

[15] I. Fonseca and G. Francfort, On the inadequacy of scaling of linear elasticity for 3D-2D asymptotics in a nonlinear setting. J. Math. Pures Appl. 80 (2001) 547-562. | MR 1831435 | Zbl 1029.35216

[16] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations with Applications to Nonlinear Continuum Physics. Springer-Verlag (to appear). | MR 2341508

[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. | MR 1617712 | Zbl 0920.49009

[18] D.D. Fox, A. Raoult and J.C. Simo, A justification of nonlinear properly invariant plate theories. Arch. Rational. Mech. Anal. 124 (1993) 157-199. | MR 1237909 | Zbl 0789.73039

[19] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses. Arch. Rational. Mech. Anal. 119 (1992) 129-143. | MR 1176362 | Zbl 0766.46016

[20] D. Kinderlehrer and P. Pedregal, Characterizations of Young mesures generated by gradients. Arch. Rational. Mech. Anal. 115 (1991) 329-365. | MR 1120852 | Zbl 0754.49020

[21] D. Kinderlehrer and P. Pedregal, Gradient Young mesures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. | MR 1274138 | Zbl 0808.46046

[22] J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions. Mathematical Institute, Technical University of Denmark, Mat-Report No. 1994-34 (1994).

[23] J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313 (1999) 653-710. | MR 1686943 | Zbl 0924.49012

[24] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549-578. | MR 1365259 | Zbl 0847.73025

[25] H. Le Dret and A. Raoult, Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Rational. Mech. Anal. 154 (2000) 101-134. | MR 1784962 | Zbl 0969.74040

[26] F.C. Liu, A Luzin type property of Sobolev functions. Indiana Univ. Math. J. 26 (1997) 645-651. | MR 450488 | Zbl 0368.46036

[27] P. Pedregal, Parametrized mesures and Variational Principles. Birkhäuser, Boston (1997). | MR 1452107 | Zbl 0879.49017

[28] E.M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press (1970). | MR 290095 | Zbl 0207.13501

[29] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, edited by R. Knops. Longman, Harlow, Pitman Res. Notes Math. Ser. 39 (1979) 136-212. | Zbl 0437.35004

[30] L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, edited by J.M. Ball. Riedel (1983). | MR 725524 | Zbl 0536.35003

[31] L. Tartar, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires. Springer-Verlag, Berlin, Lecture Notes in Phys. 195 (1994) 384-412. | MR 755737 | Zbl 0595.35012

[32] Y.C. Shu, Heterogeneous thin films of martensitic materials. Arch. Rational. Mech. Anal. 153 (2000) 39-90. | MR 1772534 | Zbl 0959.74043

[33] L.C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations. C. R. Soc. Sci. Lettres de Varsovie, Classe III 30 (1937) 212-234. | JFM 63.1064.01

[34] L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders (1969). | MR 259704 | Zbl 0177.37801

[35] W.P. Ziemer, Weakly Differentiable Functions. Sobolev spaces and functions of bounded variation. Springer-Verlag, Berlin (1989). | MR 1014685 | Zbl 0692.46022