Equi-integrability results for 3D-2D dimension reduction problems

Bocea, Marian; Fonseca, Irene

doi:10.1051/cocv:2002063

Bocea, Marian ; Fonseca, Irene

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 443-470

Résumé

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $(\nabla_{α} u_{ε} | \frac{1}{ε} \nabla_{3} u_{ε})$ bounded in $L^{p} (Ω; ℝ^{9}), 1 < p < + \infty .$ Here it is shown that, up to a subsequence, $u_{ε}$ may be decomposed as $w_{ε} + z_{ε},$ where $z_{ε}$ carries all the concentration effects, i.e. $\{{|(\nabla_{α} w_{ε} | \frac{1}{ε} \nabla_{3} w_{ε})|}^{p}\}$ is equi-integrable, and $w_{ε}$ captures the oscillatory behavior, i.e. $z_{ε} \to 0$ in measure. In addition, if ${u_{ε}}$ is a recovering sequence then $z_{ε} = z_{ε} (x_{α})$ nearby $\partial Ω .$

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/cocv:2002063

Classification : 49J45, 74B20, 74G10, 74K15, 74K35
Keywords: equi-integrability, dimension reduction, lower semicontinuity, maximal function, oscillations, concentrations, quasiconvexity

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     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},
     year = {2002},
     publisher = {EDP Sciences},
     volume = {7},
     doi = {10.1051/cocv:2002063},
     mrnumber = {1925037},
     zbl = {1044.49010},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2002063/}
}

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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

Bibliographie
Cité par

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

[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

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

[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. | Zbl | MR

[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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

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

[13] S. Conti, I. Fonseca and G. Leoni, $A$ $Γ$ -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). | Zbl | MR

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

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

[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

[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). | Zbl | MR

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

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

[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

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

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

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